Can I use decompiled source code for research?
Using a decompiler, you can work with the restored source code as good as if the authors had made the code available in the first place.


Not even close. Try running the decompiler on a free software program (to avoid any ethical tangles) to see what it generates. The output of a decompiler, especially when run on an optimized binary, is horrendous.


  Is it OK if I decompile the code to do my research?


Legality depends on the jurisdiction you are in. Ethically, my view is that there is no difference between decompiling an executable to see how it works and taking apart a mechanical device to see how it works - both are perfectly legitimate means to learn about the state of the art.


  If yes, then how do I deal with the fact that I decompiled the tool in my paper? Would it be considered rude/sketchy to say that I did it?


I don't see how it could possibly be rude. Your paper should accurately outline your methodology, so I would say you ought to include this information.


  Can I make the modified tool available to other researchers (and yes, as executables and/or source code)?


No, as it would clearly be a derivative work. As you don't have any license to use the original work, I think this would be illegal in most, if not all, jurisdictions.