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How to add a "Add a text" in a JFrame

I'm having difficulty in getting resources that will help me with an issue. I want to add a button that will let me insert text anywhere in my JFrame so I can click and add a text
Before that, I have been using JTextField and JTextArea, but these are useful in a predefined space (I believe).
What I want is like the classic Add Text button in Paint ("A" button) in which when you click on it, you will have the text cursor (like an "I") and then click somewhere in the frame so I can actually add text in that area. Which I call "adding text dynamically". Unfortunately I don't have any code to show because I'm not sure how to build the code from scratch or maybe use a method in Swing, but I'm more than keen to explain better if this is too generic for you guys.

You're going to need to be familiar with Java graphics and associated methods, particularly drawString(), so start by reading the official Graphics Tutorial.
Then learn about MouseListeners, which will help you recognize when and where the user clicks in your JFrame.
Then learn about KeyListeners, which will allow you to detect when your user is typing. You may wish to use a key bind instead, but as described, your UI is relatively simple with only one focusable component, so keylisteners are probably OK.
This general strategy should help you take a flying leap at the problem:
detect mouseclick with mouselistener
draw an insertion bar
detect keypress with keylistener
use graphics.drawstring to draw the character specified by the keypress
redraw the insertion bar
Make sure your keylistener handles the enter key, so that you know when to erase the insertion bar. You're going to paint your JFrame in between each of those steps.
The result will be crude, to say the least, and you'll notice that you have to handle a lot of issues that crop up along the way before things start to look professional. For example, you'll have to deal with font metrics. Not every character is going to be the same width, so you have to calculate the X argument to your drawString() based on the charWidth of whatever character the user types. You might do well to start with a monospaced font for that reason.

What GUI component should I use?

I am working on my homework assignment and I have to achieve the following layout. Can anyone guide me as to how to achieve the right side of the view? I have already coded the left part, it's just the right side that I don't know what to use?
Calendar GUI
Should I just use paintComponent or a JTable?

JTable doesn't seems to suit your needs. You can consider using an array of JTextArea which will be added into a JScrollPane.
In this case, you can make use of the existing behaviours from these JComponents, such as setting them as editable/non-editable. Auto scrolling for JTextArea. JTextArea also allows displaying of multiple lines of records.
Should I just use paintComponent
I supposed you meant by custom painting. Using custom painting will give you alot of freedom to do your own customizations, however if the current JComponents are able to fulfill you needs, then I think you shouldn't try to reinvent the wheel, especially when you need to deal with printing text. Aligning the text properly in custom painting could cost you alot more time than learning how to use various existing JComponents.

How can I change colors of components when the mouse is pressed in a JFrame in Java?

I am coding a piano in java using rectangles from the java.awt.graphics library. I am using a mouselistener to play the sound of each individual key when the mouse clicks a certain area on the JFrame.
How would I add a shape to the panel upon clicking, repaint, then repaint the keyboard back over top when the user releases the mouse?

Consider adding JLabels to a JPanel that uses GridLayout. Give each JLabel a MouseListener and either swap ImageIcons on mousePress/mouseRelease or change the JLabel's background with press and release. If you go the latter route, you'll want to make sure that the JLabels opaque property is set to true so that the background colors show.
Then for the black keys, you can add the above JPanel to a JLayeredPane and on top of this, add another JPanel that holds the black keys that function in the same way.
Also, you'll want to take care to "play" any notes in a background thread, such as can be obtained with a SwingWorker so as not to tie up the Swing event thread and completely freeze your program.

Consider solution: source
It might not be exactly what you're after, but it might give you an idea of how to approach your problem. It took me a long time to figure out how to use JLayeredPane without setting a null layout, but in the end this was the best I could come up with. Also, assumed some naming conventions for your sound files. :p

Handle overlapping components in java swing

I am working on an app which when used for loading a file and drawing the contained components, may result into painting of overlapping components. For example, consider a big rectangle box containing text line inside it.
Now because these components are overlapping, it is difficult for the user to select the inner text box in this case as it has been overlapped by the rectangle box.
We were thinking of solving this with allowing the users to actually move any component to a layer below the current one. But this has its own limitations on the usability side, as then for every such case the user will have to move the bigger or the most recently painted component to a layer below and then do the other processing on the inner components like dragging etc. There can be more than 2 components at the same 2d (x & y position) in this app.
I am sure that there should be a better solution for this and could someone please provide some pointers on the implemention part of it.

I am not sure, whether i understand your question. Do you have problems, with the action listeners or because of drawing the components?
This How to Use Root Panes may be helps.
You can create a Glasspane in Java

Here's a simple example of an object drawing program that supports click and drag for multiple selections. More advanced programs often implement z-order functionality (Move Forward, Move Backward, Move to Front, Move to Back) by simply re-ordering the node list.
As an alternative to the layered pane approach, also consider JInternalFrame. This example shows How to Use Actions to manage a menu from which hidden windows may be brought to the fore.

Java Swing group of Jbuttons without raised portion

I wanted to know how to display a group of JButtons to look like smooth panel without raised portion.
thanks

button.setBorder(null);
You may want to look at some of the other "setXXX" method that control painting as well.

I've often just used standard JLabels and added mouseListeners to make them clickable. Alternatively, you could get more advanced and create your own ButtonUI class if you want to really fine-grained control over the rendering of the buttons.

If you want the buttons to be in a row, you can put them in a JToolBar and set Rollover to true. This will make flat buttons that, with mouse over, look raised.