I can only answer two of your questions:

• How to know if a steam game is “deck verified”? There is a “Steamdeck compatability” box for every game on steam that will give on one of four statuses telling you supported/partially supported/not supported/unknown Image
  • That being said, steam is a bit conservative about “supported” games. ProtonDB is a more complete list of compatible games and gives guidance on how to play games steam considers “unsupported”.
• Can you use steam input for non-steam games? Games must be added/launched through steam for the steam input settings to work, so in most cases you should be able to set it up by adding the “non-steam” game to steam.