I'm trying to find the name of the file I'm editing inside of vim.
For the current buffer this will give you name of the file,
echo "You're editing " bufname("%")
(just put it in some file.vim, and source it ":so %". But I don't think this is what you need.
So I can use it to map F5 to compile this file, for testing purposes. Would of course be great if I could recognize the file format, and choose compiler accordingly, but realy not necessary. If I find the name of the file, I could do that myself. But I really can't find any way to get the name of the file I'm editing.
You could do several things. If vim recognizes the filetype you're editing, you could map F5 to your compiler command and put that command in its specific ftplugin directory, so it will be valid only for that filetype. For example, you put
nmap <f5> :!compilername %<cr>
in cpp.vim, fortran.vim, python.vim ... (of course not the same command, you're not going to compile fortran with cpp compiler, you put fortrancompiler % in that case)
and that will be like you did in command prompt compilername file.cpp (the % gives the name of the current file you're editing)
That is one option. You could also try according to extension set autocmd commands in your vimrc so it recognizes the filetype according to extension of the file. This is all standard usage of vim, nothing uncommon. Now, I'm not sure how you'd like to go about this, so I'll just point you to Vim Wiki where you can find all kinds of articles that cover this (and also a few tips for some compilers).
I know of the :make command, and have already mapped that, but for small scripts/testing programs, I really don't want to first have to write a simple makefile.
Yes, that seems like overkill. Personally (when I'm on Win), I find simple batch file much easier to write for most things except big projects. If the program is in one or two files, I just compile it using some oneliner mapping.