You can select to run them in proton, rather than just normal wine, in lutris. I’ve been doing it to run games from gog for years with few issues, namely games that are old and have the same problems working on windows as well. So… Yes they will?

Linux compatibility is fine. You can pretty easily install gog games by logging into your gof account on lutris. It’ll let you see your whole library and install directly from it, and any games that work in proton will work just fine using proton via lutris. You can also just add your gog game as a non-steam game and run it there.