Yes. The Flip 5 has a different cover screen than the 3. It’s significantly larger. Just look up a a photo comparison of how significantly different they are. Yes, you can run literally any app or APK file from the cover screen on the Flip 5. The ones I use most are Discord, Spotify, YouTube, Google Messages, Voyager, my notes app, and Red Reader (Reddit).
Sometimes they can be a bit buggy or don’t function very well (esp games), so I don’t like using certain apps on the cover screen.
The only caveat is you need to install a free app by Samsung called Good Lock from the Galaxy Store first. Evidently it is not available in certain regions. In these regions, then yes, the cover screen functionality is locked down to only a few approved apps.
I just have my mail set to silent notifications so I still see if I got an email but it doesn’t interrupt anything.