File extensions are really nothing to Linux.
Data files can be opened as long as there's a program that can understand the format. Since there are virtually millions of file formats out there, we would have to concentrate on a few of them, and see if there are visors/editors for those files. Most commons graphics and sound formats can be opened without problems. Some restricted formats might be problematic, like drm'ed media player files (I don't have any interest in using drm'ed media anyway).
Exe files are not data files. They are programs. And windows programs can't be run in linux natively because they are two different OSes from the root, they are plainly not compatible. Nonetheless, you can try to run *some* windows programs under wine, which is a reimplementation of the Windows API on top of linux and xlib. Whether a concrete program will work (or how if at all) depends entirely on the program itself. You can always check the wine database:
This has nothing to do with the file extensions however. You could name a linux program "foo.exe", but the internal format of the file would continue to be this of an ELF executable for linux, and not a Windows exe file.