If you're wanting something similar to SteamOS right now that has much wider hardware support for your gaming handhelds like the Lenovo Legion Go, ASUS ROG Ally, GPD, Ayaneo or even your laptop / desktop then Bazzite is likely one of the best choices.
Wouldn’t a DP - HDMI cable work?
(Don’t know anything about this)
There’s pretty much a single DP -> HDMI adapter that supports VRR by Cable Matters. Officially I don’t think VRR should work over adapters, or at least other manufacturers don’t invest the time to make it work.