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What is the best way to create a test automation suite as a service

I am trying to create the following setup :
A Selenium (Java) project that has a set of 10 automated test cases.
When this project is executed, it generates an HTML test execution report.
This project should be 'hosted' on an internal network.
Anyone who has access to the network should be able to 'invoke' this project, which in turn executes the test cases and passes the HTML report to the person who invoked it.
The project should be accessible ONLY for execution and the code should NOT be accessible.
My goal is that this implementation should be executable by any framework irrespective of the technology that the framework uses. I was thinking of creating the project as a WebService using Java (servlet).
My question is:
Can this implementation be accessed by any external automation framework ?
Are there any limitations to this implementation?
Is there a better way to implement this requirement?
Thanks in advance.

You can create a maven project and have your automated tests under maven test folder.Configure your tests to run through POM.xml(use maven surefire plugin).Configure a jenkins job to run the maven test.Anybody with access the jenkins can build/run this task!
Below link should give you a headstart
http://learn-automation.com/selenium-integration-with-jenkins/

As a matter of fact, it is something we did on one of our projects. As I cannot share specifics, I will give you overall architectural view of the project.
The core of all things was a service that could run JUnit tests on requests. It was a Soap web-service, but nothing stops you from making it REST. To implement this you need to implement your version of JUnit test runners (see for example: http://www.mscharhag.com/java/understanding-junits-runner-architecture or https://github.com/junit-team/junit4/wiki/Test-runners)
If you use JUnit as test framework for running your Selenuim tests this may be a great solution for you - JUnit will generate HTML reports for you if you configure it properly, it will hide actual test suite implementation from users and run test suite on demand. This solution is also great because it operates on JUnit level and does not care about what kind of tests it actually runs, so it can be also reused for any other kind of automated tests.
So to answer all your questions:
Can this implementation be accessed by any external automation
framework ? -> yes, it can be accessed by anybody who able send http
requests
Are there any limitations to this implementation? -> none that I am
aware of
Is there a better way to implement this requirement? -> well, I
didn't actually work with TestNG much so I don't know if it is
easier or more difficult to do it on Junit level. You can use
Jenkins or other CI tool as well to achieve same results - they can
run JUnit tests for you and almost always have API ready for this,
although those APIs may be not perfect.
So I'd say that if you need this only for one thing you can use CI tools for this purpose, if you don't have CI tools available then choice has been made for you. However, from our experience, having this kind of service was a great asset for a company and I really wonder why there's no such products available elsewhere yet.

How to integrate JUnit into already developed application?

I'm very much new to JUnit. We want to integrate JUnit into our old and big Enterprise Java application(which has many projects associated with it) developed long back.We want to do it without touching java files and on framework level. Is it possible? If yes, please share me the links or information on how to do it?

I can't comment (yet) so here is my recommendation as an answer:
"Working Effectively with Legacy Code" by Michael Feathers covers all scenarios of testing / maintaining etc. of old, huge applications in a very readable way.

Why would you touch existing Java files for writing unit tests ( if you are ready to leave non-testable classes in your source as they are) ?
Isn't integrate JUnit means writing unit tests for existing classes using JUnit ( for which unit tests have not been written so far) OR is there any other meaning you intend too?.
We keep test classes written using JUnit in a separate source-folder so those don't mess with your existing code.
I don't see any concern here. Just add JUnit jar into your project's build path ( by including jar OR by adding maven dependency OR by adding Gradle dependency ) and start writing unit tests for your testable classes and you are done with your integration.
You might choose to not include source-folder-for-tests into your deployment build.
Sometimes in your old code, some classes might not be testable so you will have to tweak those a little if wish to cover those too.
Hope it helps !!

Where to learn about writing test cases?

I'm a CS master student. Throughout my studies I coded many course projects in Java. Soon I will graduate. When I explore some github projects I often find people organize their projects as /main and /test. I have never organized it in such a way, i.e. I always have my source code files without any test directories. I think that folder often contains what I think is called 'test cases' or so.
Since I will find a job soon, then I would like to learn about production-quality code.
My questions:
Why people often have that folder? What does it contain?
Can you provide me with a link to a good tutorial about the practice of testing in java? i.e how to do it? In a nutshell I wanna understand the idea of that /tests/ folder.

I often find people organize their projects as /main and /test
This is a matter of taste. Not 100% sure but at least maven projects have such organization.
From Maven: Introduction to the Standard Directory Layout, this would be the project layout:
src
main
java <-- your Java source code
resources
filters
config
scripts
webapp
test
java <-- your unit tests for Java
resources
filters
it
assembly
site
Why people often have that folder? What does it contain?
Usually, people write test cases to cover the code and check if the code works as expected. This is known as Code Coverage. Code coverage also serves as regression tests in case somebody makes changes in the code for enhancements like code refactoring.
The test cases you will find them usually are for Unit Testing. Depending on the type of the project, you could also find Integration Tests.
There is also Test Driven Development, or TDD, which is a practice whose basis is writing the test cases before writing the real code.
Can you provide me with a link to a good tutorial about the practice of testing in java?
This is off topic for the site. There are plenty tutorials on the net about this.

I don't have a separate folder for mine but usually people keep their Unit Tests in that folder. A unit test generally sets up "fake" data to test a given class so that a developer can easily debug any issues.

The reason people provide a /test folder is to contain unit test for their project.
There are really many ways of testing Java but JUnit is a very commonly used method of testing.

It is a good practice to write tests for your code. Begin with writing Unit Tests. I found this tutorial very useful. Writing test ensures that your code behaves as expected , corner cases are tested and adding new code in the future does not break existing functionality.
There are also mocking frameworks like JMock and Mockito that make writing stubs and drivers for your methods easy.
What is even more interesting is people prefer writing tests before they write the actual implementation. This approach is called Test Driven Development or Extreme Programming. Writing tests first ensure one already has a prep code or pseudo code for the methods in mind.

UnitTest autorunner on Intellij IDEA for java

I try to use plugin 'fireworks' http://plugins.jetbrains.com/plugin?pr=idea&pluginId=1106 for autrorunning unittests for java code. but it doesn't work. Is any alternative for it?
Is anybody autroexecute unittests for java code after sources have changed?
PS
My project is tiny - it's just exercise for Algorithms courses so i don't want build complex deployment system on Maven or something to resolve so trivial tasks. So if any plugin for IDEA has exist it will be better for me.
But if you have simple decision on Maven it also can fit to it.

You could bind the ^S to run the tests (^shiftR) that would pretty much give you what you are after(I think).
Then Set the test runner to always save before running.
Then every time you do a ^S the tests will be triggered.

Yes Infinitest also suits this purpose. For larger projects it becomes annoying but for your small project it should work well. Also, it works well in Eclipse.

Regarding generation of junit test cases of java classes

I am pure developer and spent most of time in writing the java classes , but the case of junit test cases , I become little bit lazy is there any plugin that can be installed in eclipse that will generate my junit test cases automatically..!! Please advise

MoreUnit is an awesome plugin that generates test classes and allows easy navigation between test and class under test. It can also generate test method stubs and mock boilerplate, but I don't think these are as useful.
As for the test code itself, the only way around writing it is to get someone else to write it (i.e. executable specifications).

The JUnit plugin stubs out the methods for you, which is all you should need. It's up to you to know what and how to test. There is this tool from Parasoft, but I have never used it, so can't say how it is.