When I bought Heresphere on Steam, can I also use this on the SteamFrame (soon(tm)) too, when the SteamFrame is released, or will the SteamFrame require a other version?

You can stream the PCVR version of HereSphere on the SteamFrame. The standalone version for the SteamFrame still needs to be developed.

I understood, that the SteamFrame should be able to run games and applications like a SteamDeck would, natively. This would mean, that I could play Dredge (without it needing a VR version), that would go for HereSphere too, so that I can watch movies from my Plex?

That would also be cool, if there was a Plex integration... Anyone know if there is something like that? Quick search resulted in No Plex-HereSphere Integration.