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Why Char is actually a NumericType in Java, but not a SymbolicType or String?

Regarding Java syntax, there is a NumericType which consists of IntegralType and FloatingPointType. IntegralTypes are byte, short, int, long and char.
At the same time, I can assign a single character to char variable.
char c1 = 10;
char c2 = 'c';
So here is my question. Why char in numeric type and how JVM convert 'c' to a number?

Why char in numeric type...
Using numbers to represent characters as indexes into a table is the standard way the text is handled in computers. It's called character encoding and has a long history, going back at least to telegraphs. For a long time personal computers used ASCII (a 7-bit encoding = 127 characters plus nul) and then "extended ASCII" (an 8-bit encoding of various forms where the "upper" 128 characters had a variety of interpretations), but these are now obsolete and suitable only for niche purposes thanks to their limited character set. Before personal computers, popular ones were EBCDIC and its precursor BCD. Modern systems use Unicode (usually by storing one or more of its transformations such as UTF-8 or UTF-16) or various standardized "code pages" such as Windows-1252 or ISO-8859-1.
...and how JVM convert 'c' to a number?
Java's numeric char values map to and from characters via Unicode (which is how the JVM knows that 'c' is the value 0x0063, or that 'é' is 0x00E9). Specifically, a char value maps to a Unicode code point and strings are sequences of code points.
There's quite a lot about the char data type, including why the value is 16 bits wide, in the JavaDoc of the Character class:
Unicode Character Representations
The char data type (and therefore the value that a Character object encapsulates) are based on the original Unicode specification, which defined characters as fixed-width 16-bit entities. The Unicode Standard has since been changed to allow for characters whose representation requires more than 16 bits. The range of legal code points is now U+0000 to U+10FFFF, known as Unicode scalar value. (Refer to the definition of the U+n notation in the Unicode Standard.)
The set of characters from U+0000 to U+FFFF is sometimes referred to as the Basic Multilingual Plane (BMP). Characters whose code points are greater than U+FFFF are called supplementary characters. The Java platform uses the UTF-16 representation in char arrays and in the String and StringBuffer classes. In this representation, supplementary characters are represented as a pair of char values, the first from the high-surrogates range, (\uD800-\uDBFF), the second from the low-surrogates range (\uDC00-\uDFFF).
A char value, therefore, represents Basic Multilingual Plane (BMP) code points, including the surrogate code points, or code units of the UTF-16 encoding. An int value represents all Unicode code points, including supplementary code points. The lower (least significant) 21 bits of int are used to represent Unicode code points and the upper (most significant) 11 bits must be zero. Unless otherwise specified, the behavior with respect to supplementary characters and surrogate char values is as follows:
The methods that only accept a char value cannot support supplementary characters. They treat char values from the surrogate ranges as undefined characters. For example, Character.isLetter('\uD840') returns false, even though this specific value if followed by any low-surrogate value in a string would represent a letter.
The methods that accept an int value support all Unicode characters, including supplementary characters. For example, Character.isLetter(0x2F81A) returns true because the code point value represents a letter (a CJK ideograph).
In the Java SE API documentation, Unicode code point is used for character values in the range between U+0000 and U+10FFFF, and Unicode code unit is used for 16-bit char values that are code units of the UTF-16 encoding. For more information on Unicode terminology, refer to the Unicode Glossary.

Because underneath Java represents chars as Unicode. There is some convenience to this, for example you can run a loop from 'A' to 'Z' and do something. It's important to realize, however, that in Java Strings aren't strictly arrays of characters like they are in some other languages. More info here

Internally char is stored as ASCII (or UNICODE) code which is integer. The difference is in how it is processed after it is read from memory.
In C/C++ char and int are very close and is type casted implicitly. Similar behavior in Java shows the relation between C/C++ and Java as JVM is written in C/C++.

Besides being able to do arithmetic operations on chars which sometimes comes handy (like c >= 'a' && c <= 'z') I would say it is a design decision driven by the similar approach taken in other languages when Java was invented (primarily C and C++).
The fact that Character does not extend Number (as other numeric primitive wrappers do) somehow indicates that Java designers tried to find some kind of a compromise between numeric and non-numeric nature of characters.
DISCLAIMER I was not able to find any official docs about this.

4 byte unicode character in Java

I am writing unit tests for my custom StringDatatype, and I need to write down 4 byte unicode character.
"\U" - not working (illegal escape character error)
for example: U+1F701 (0xf0 0x9f 0x9c 0x81). How it can be written in a string?

A Unicode code point is not 4 bytes; it is an integer (ranging, at the moment, from U+0000 to U+10FFFF).
Your 4 bytes are (wild guess) its UTF-8 encoding version (edit: I was right).
You need to do this:
final char[] chars = Character.toChars(0x1F701);
final String s = new String(chars);
final byte[] asBytes = s.getBytes(StandardCharsets.UTF_8);
When Java was created, Unicode did not define code points outside the BMP (ie, U+0000 to U+FFFF), which is the reason why a char is only 16 bits long (well, OK, this is only a guess, but I think I'm not far off the mark here); since then, well, it had to adapt... And code points outside the BMP need two chars (a leading surrogate and a trailing surrogate -- Java calls these a high and low surrogate respectively). There is no character literal in Java allowing to enter code points outside the BMP directly.
Given that a char is, in fact, a UTF-16 code unit and that there are string literals for these, you can input this "character" in a String as "\uD83D\uDF01" -- or directly as the symbol if your computing environment has support for it.
See also the CharsetDecoder and CharsetEncoder classes.
See also String.codePointCount(), and, since Java 8, String.codePoints() (inherited from CharSequence).

String s = "𩸽";
Technically this is one character. But be careful s.length() will returns 2. Also java won't compile String s = '𩸽'. Java don't promise you that String.length() shall returns exact number of characters, it returns just number of java-chars required for store this string.
Real number of characters can be obtained from s.codePointCount(0, s.length()).

jshell> String s = "🏳";
s ==> "🏳️"
jshell> s.codePointCount(0, s.length());
$5 ==> 2

Obtain ascii values of chars in C#

I am accustomed to coding in java, but recently I have been making some ASP webpages that use C#.
In Java chars are default represented by their numeric ascii value unless you put them with a string. I have been unable to repeat this in C#.
What do I need to do to get ascii values of chars in C#?

char in .Net is a 2-byte structure representing a UTF-16 encoding of a unicode code point - of which ASCII is a tiny subset. But some unicode code points including certain Kanji characters require more than two bytes, and these are represented in a .Net string as a surrogate pair. Thus the most general way to get an unicode code point value for a character in a string at a specified index is Char.ConvertToUtf32(string s, int index)
For instance, the following enumerates the unicode code point values in a string:
public static IEnumerable<int> Utf32CodePoints(string s, int index)
{
for (int length = s.Length; index < length; index++)
{
yield return char.ConvertToUtf32(s, index);
if (char.IsSurrogatePair(s, index))
index++;
}
}
If you explicitly want only ASCII values and want to skip non-ASCII characters, you could use the ASCII decoder with appropriate exceptions, as shown here: Encoding.ASCII Property. Alternatively, just cast each char to an int and check if its value falls between U+0000 and U+007F, which is the defined range for ASCII.

ASCII is very small subset of characters that can be represented in C#/Java.
Fastest way to get ASCII code (assuming you know that value fits in ASCII range):
var ascii = ((int)c) & 0x7F;
You may want to add range checks (0-0x7F) and fail if value falls outside the range. Alternatively you can use Encoding.ASCII to do conversion (will replace characters outside of the range with question marks).
Note: if your "ascii" actually mean "numeric value"/UTF-16 Unicode code than basic cast to ushort (or int) will work:
var code = (int)c;

char to Unicode more than U+FFFF in java?

How can I display a Unicode Character above U+FFFF using char in Java?
I need something like this (if it were valid):
char u = '\u+10FFFF';

You can't do it with a single char (which holds a UTF-16 code unit), but you can use a String:
// This represents U+10FFFF
String x = "\udbff\udfff";
Alternatively:
String y = new StringBuilder().appendCodePoint(0x10ffff).toString();
That is a surrogate pair (two UTF-16 code units which combine to form a single Unicode code point beyond the Basic Multilingual Plane). Of course, you need whatever's going to display your data to cope with it too...

Instead of using StringBuilder, you can also use a function
directly found in the Character class. The function is
toChars() and it has the following spec:
Converts the specified character (Unicode code point) to
its UTF-16 representation stored in a char array.
So you don't need to exactly know how the surrogate pairs look
like and you can directly use the code point. An example code
then looks as follows:
int ch = 0x10FFFF;
String s = new String(Character.toChars(ch));
Note that the datatype for the code point is int and not char.

Unicode characters can take more than two bytes which can't be in general hold in a char.

Source
The char data type are based on the original Unicode specification, which defined characters as fixed-width 16-bit entities. The range of legal code points is now U+0000 to U+10FFFF, known as Unicode scalar value.
The set of characters from U+0000 to U+FFFF is sometimes referred to as the Basic Multilingual Plane (BMP). Characters whose code points are greater than U+FFFF are called supplementary characters. The Java 2 platform uses the UTF-16 representation in char arrays and in the String and StringBuffer classes. In this representation, supplementary characters are represented as a pair of char values, the first from the high-surrogates range, (\uD800-\uDBFF), the second from the low-surrogates range (\uDC00-\uDFFF).
A char value, therefore, represents Basic Multilingual Plane (BMP) code points, including the surrogate code points, or code units of the UTF-16 encoding. An int value represents all Unicode code points, including supplementary code points. The lower (least significant) 21 bits of int are used to represent Unicode code points and the upper (most significant) 11 bits must be zero. Unless otherwise specified, the behavior with respect to supplementary characters and surrogate char values is as follows:
The methods that only accept a char value cannot support supplementary characters. They treat char values from the surrogate ranges as undefined characters. For example, Character.isLetter('\uD840') returns false, even though this specific value if followed by any low-surrogate value in a string would represent a letter.
The methods that accept an int value support all Unicode characters, including supplementary characters. For example, Character.isLetter(0x2F81A) returns true because the code point value represents a letter (a CJK ideograph).
In the J2SE API documentation, Unicode code point is used for character values in the range between U+0000 and U+10FFFF, and Unicode code unit is used for 16-bit char values that are code units of the UTF-16 encoding.

Java Unicode encoding

A Java char is 2 bytes (max size of 65,536) but there are 95,221 Unicode characters. Does this mean that you can't handle certain Unicode characters in a Java application?
Does this boil down to what character encoding you are using?

You can handle them all if you're careful enough.
Java's char is a UTF-16 code unit. For characters with code-point > 0xFFFF it will be encoded with 2 chars (a surrogate pair).
See http://www.oracle.com/us/technologies/java/supplementary-142654.html for how to handle those characters in Java.
(BTW, in Unicode 5.2 there are 107,154 assigned characters out of 1,114,112 slots.)

Java uses UTF-16. A single Java char can only represent characters from the basic multilingual plane. Other characters have to be represented by a surrogate pair of two chars. This is reflected by API methods such as String.codePointAt().
And yes, this means that a lot of Java code will break in one way or another when used with characters outside the basic multilingual plane.

To add to the other answers, some points to remember:
A Java char takes always 16 bits.
A Unicode character, when encoded as UTF-16, takes "almost always" (not always) 16 bits: that's because there are more than 64K unicode characters. Hence, a Java char is NOT a Unicode character (though "almost always" is).
"Almost always", above, means the 64K first code points of Unicode, range 0x0000 to 0xFFFF (BMP), which take 16 bits in the UTF-16 encoding.
A non-BMP ("rare") Unicode character is represented as two Java chars (surrogate representation). This applies also to the literal representation as a string: For example, the character U+20000 is written as "\uD840\uDC00".
Corolary: string.length() returns the number of java chars, not of Unicode chars. A string that has just one "rare" unicode character (eg U+20000) would return length() = 2 . Same consideration applies to any method that deals with char-sequences.
Java has little intelligence for dealing with non-BMP unicode characters as a whole. There are some utility methods that treat characters as code-points, represented as ints eg: Character.isLetter(int ch). Those are the real fully-Unicode methods.

You said:
A Java char is 2 bytes (max size of 65,536) but there are 95,221 Unicode characters.
Unicode grows
Actually, the inventory of characters defined in Unicode has grown dramatically. Unicode continues to grow — and not just because of emojis.
143,859 characters in Unicode 13 (Java 15, release notes)
137,994 characters in Unicode 12.1 (Java 13 & 14)
136,755 characters in Unicode 10 (Java 11 & 12)
120,737 characters in Unicode 8 (Java 9)
110,182 characters in Unicode 6.2 (Java 8)
109,449 characters in Unicode 6.0 (Java 7)
96,447 characters in Unicode 4.0 (Java 5 & 6)
49,259 characters in Unicode 3.0 (Java 1.4)
38,952 characters in Unicode 2.1 (Java 1.1.7)
38,950 characters in Unicode 2.0 (Java 1.1)
34,233 characters in Unicode 1.1.5 (Java 1.0)
char is legacy
The char type is long outmoded, now legacy.
Use code point numbers
Instead, you should be working with code point numbers.
You asked:
Does this mean that you can't handle certain Unicode characters in a Java application?
The char type can address less than half of today's Unicode characters.
To represent any Unicode character, use code point numbers. Never use char.
Every character in Unicode is assigned a code point number. These range over a million, from 0 to 1,114,112. Doing the math when comparing to the numbers listed above, this means most of the numbers in that range have not yet been assigned to a character yet. Some of those numbers are reserved as Private Use Areas and will never be assigned.
The String class has gained methods for working with code point numbers, as did the Character class.
Get the code point number for any character in a string, by zero-based index number. Here we get 97 for the letter a.
int codePoint = "Cat".codePointAt( 1 ) ; // 97 = 'a', hex U+0061, LATIN SMALL LETTER A.
For the more general CharSequence rather than String, use Character.codePointAt.
We can get the Unicode name for a code point number.
String name = Character.getName( 97 ) ; // letter `a`
LATIN SMALL LETTER A
We can get a stream of the code point numbers of all the characters in a string.
IntStream codePointsStream = "Cat".codePoints() ;
We can turn that into a List of Integer objects. See How do I convert a Java 8 IntStream to a List?.
List< Integer > codePointsList = codePointsStream.boxed().collect( Collectors.toList() ) ;
Any code point number can be changed into a String of a single character by calling Character.toString.
String s = Character.toString( 97 ) ; // 97 is `a`, LATIN SMALL LETTER A.
a
We can produce a String object from an IntStream of code point numbers. See Make a string from an IntStream of code point numbers?.
IntStream intStream = IntStream.of( 67 , 97 , 116 , 32 , 128_008 ); // 32 = SPACE, 128,008 = CAT (emoji).
String output =
intStream
.collect( // Collect the results of processing each code point.
StringBuilder :: new , // Supplier<R> supplier
StringBuilder :: appendCodePoint , // ObjIntConsumer<R> accumulator
StringBuilder :: append // BiConsumer<R,​R> combiner
) // Returns a `CharSequence` object.
.toString(); // If you would rather have a `String` than `CharSequence`, call `toString`.
Cat 🐈
You asked:
Does this boil down to what character encoding you are using?
Internally, a String in Java is always using UTF-16.
You only use other character encoding when importing or exporting text in or out of Java strings.
So, to answer your question, no, character encoding is not directly related here. Once you get your text into a Java String, it is in UTF-16 encoding and can therefore contain any Unicode character. Of course, to see that character, you must be using a font with a glyph defined for that particular character.
When exporting text from Java strings, if you specify a legacy character encoding that cannot represent some of the Unicode characters used in your text, you will have a problem. So use a modern character encoding, which nowadays means UTF-8 as UTF-16 is now considered harmful.

Have a look at the Unicode 4.0 support in J2SE 1.5 article to learn more about the tricks invented by Sun to provide support for all Unicode 4.0 code points.
In summary, you'll find the following changes for Unicode 4.0 in Java 1.5:
char is a UTF-16 code unit, not a code point
new low-level APIs use an int to represent a Unicode code point
high level APIs have been updated to understand surrogate pairs
a preference towards char sequence APIs instead of char based methods
Since Java doesn't have 32 bit chars, I'll let you judge if we can call this good Unicode support.

Here's Oracle's documentation on Unicode Character Representations. Or, if you prefer, a more thorough documentation here.
The char data type (and therefore the value that a Character object
encapsulates) are based on the original Unicode specification, which
defined characters as fixed-width 16-bit entities. The Unicode
standard has since been changed to allow for characters whose
representation requires more than 16 bits. The range of legal code
points is now U+0000 to U+10FFFF, known as Unicode scalar value.
(Refer to the definition of the U+n notation in the Unicode standard.)
The set of characters from U+0000 to U+FFFF is sometimes referred to
as the Basic Multilingual Plane (BMP). Characters whose code points
are greater than U+FFFF are called supplementary characters. The Java
2 platform uses the UTF-16 representation in char arrays and in the
String and StringBuffer classes. In this representation, supplementary
characters are represented as a pair of char values, the first from
the high-surrogates range, (\uD800-\uDBFF), the second from the
low-surrogates range (\uDC00-\uDFFF).
A char value, therefore, represents Basic Multilingual Plane (BMP)
code points, including the surrogate code points, or code units of the
UTF-16 encoding. An int value represents all Unicode code points,
including supplementary code points. The lower (least significant) 21
bits of int are used to represent Unicode code points and the upper
(most significant) 11 bits must be zero. Unless otherwise specified,
the behavior with respect to supplementary characters and surrogate
char values is as follows:
The methods that only accept a char value cannot support supplementary characters. They treat char values from the surrogate
ranges as undefined characters. For example,
Character.isLetter('\uD840') returns false, even though this specific
value if followed by any low-surrogate value in a string would
represent a letter.
The methods that accept an int value support all Unicode characters, including supplementary characters. For example,
Character.isLetter(0x2F81A) returns true because the code point value
represents a letter (a CJK ideograph).

From the OpenJDK7 documentation for String:
A String represents a string in the
UTF-16 format in which supplementary
characters are represented by
surrogate pairs (see the section
Unicode Character Representations in
the Character class for more
information). Index values refer to
char code units, so a supplementary
character uses two positions in a
String.