Did you open and play back a video first? If you just open QuickTime Player and don't interact with it, it does stay open if you switch apps. If you open a video, then close it, the app closes if you switch to a different app. I just tried it again and that's how it behaved. (I'm running Catalina still, so they may have fixed it in Big Sur.)
No idea. I've seen it behave this way for at least the past year. I just tried it on a different Mac and it behaves similarly, except this other Mac causes QT to open up the File Open dialog immediately on launch. (Version 10.5 of QuickTime player for me too. macOS 10.15.7 though.)
