I use a cheap USB-C hub that automatically outputs video to my TV independently of when any cable is plugged. I don’t know why that’s happening to you.
I also have a 8bitdo SN30 PRO. I use it in Switch mode and didn’t have any problems with it. In Switch mode you have Gyro, which I believe X-Input doesn’t have (someone else may confirm). Since the games will be receiving Steam Input I don’t think it matters that much which mode you choose.
Yes, it should. Even when I was using Windows 11 the video output going to the TV was automatic. The audio was messed though. I had to either restart the Deck while already plugged or Troubleshoot the audio every time.
Did you change any video configuration regarding multiple monitors either on Desktop Mode or Game Mode? Have you checked that?