For me, koala went a bit backwards with the his answers, IMO, because he only talked about window capture while completely missing the remaining questions about game capture and display capture. So I will try to explain it from my own experience instead.
1) If you're capturing a game, Game Capture is generally the best option of course.
Performance wise, from Windows 10 onwards, they seem to be similar, though it likely that Display Capture is "slowest" of them all. Window Capture should have same performance as Game Capture though.
2) There shouldn't be any performance impacts between those two modes in fullscreen. But, then again, it's OBS, so sometimes weird things can happen...
3) It all depends on what you're trying to capture.
If you are trying to have a game in your OBS capture, Game Capture is the way to go, generally speaking. If there are issues, the fallback is Window Capture (though it will capture the game with all overlays like Steam and you cannot hide them like in Game Capture). Display Capture is last resort option for that, if Window Capture also fails.
Window Capture is the main choice if you're capturing something other than a game, like a browser window or any application window. As koala already mentioned, it has some options, but Automatic capture method will probably do the job well enough.
Display Capture should generally be used as a last-resort option, or if you are gaming on a separate monitor and it's somewhat empty - as it always shows everything that is on that screen (saves some "black void" moments, if done correctly).
Preferentially, I would say it goes "Game Capture" >= "Window Capture" > "Display Capture", or "Window Capture" > "Display Capture" if you're capturing something not-gaming-related.
4) If we're specifically talking about Windows 11, there shouldn't be any big difference since it didn't change too much from Windows 10. HOWEVER, at the time I'm answering, it seems that OBS might be suffering from a "0 kbps" stream issue on some Windows 11 machines (with increasing numbers nowadays it seems), while Windows 10 machines seem to be unaffected by that. So, the correct answer is probably "yesn't" (both yes and no).