Recently I took GPL'd source code, simplify them, remove few functions that I don't need, add new functions, and use them in my closed source program as plug-in. Is that legally allowed?
Secondly, I want to release the new source code as MIT license, is that allowed?
Secondly, I want to release the new source code as MIT license, is that allowed?
Your work is a derivative work of the original copyrighted source code.
The GPL only gives permission for you to distribute such a derivative work if it is also GPL'd. Thus, if you're only using your code privately, you should be OK. If you actually distribute your closed-source derivative work to others, then you have reused the original copyrighted source code in a manner not approved by the rightsholder's license. You are then violating someone else's copyright.
This applies whether you distribute your derivative code in a closed-source product, or under a non-GPL open-source license (like the MIT license).
(I am not a lawyer but this is pretty basic GPL interpretation.)
Edit: to clarify private use vs. 'distribution' distinction.