The Deck can output up to 4K 60Hz with the right dock, so the picture quality is not going to be limited by the supported resolution of the Deck. What will limit the picture quality is that SteamOS by default runs games at 1280x800 or 1280x720 for 16:9 external screens, regardless of the actual selected resolution. It will upscale games rendered at 720p to whatever the actual output resolution (1080p or 4K) is. There is an option in the per-game settings in the SteamOS UI to set the resolution for each game. If you pick Native, it will allow the game to render up to the screen's native resolution for a full-quality image, no different than you would get on a normal PC. However, the Steam Deck's GPU isn't very powerful compared to a desktop PC so you aren't going to be able to push most games that high. A lot of older titles and 2D games can run fine at native 1080p on the Deck though.

As long as you render the games at the Steam Deck native resolution (i.e. 1280x800) and use the FSR upscaling option this works perfectly. I do this to output to my 4K TV and the games run more or less the same as they would on the Deck itself with great visual quality. A 1080p TV should work the same if not better.

Make sure to put your TV in "gaming mode" or whatever it's called in yours. Otherwise it'll do some extra calculations to improve image quality which will cause noticable input lag. I've struggled with that issue for a while, thinking the Deck itself caused the input lag when displaying on an external display, but it actually was the TV which caused it. After setting that mode, together with FSR, it was basically the same performance as in handheld mode from what I can tell so far.

I use mine docked fairly frequently. No major issues that I've had. Just note that if you change the games resolution to higher than the steam decks native screen, you'll sacrifice framerate for it.

It depends on your dock, your TV and whatever you connect your SteamDeck to. I tried connecting my SteamDeck to my Yamaha Receiver/AMP and it got confused by the HDMI CEC, and so I had to buy an HDMI CEC filter adapter so that it doesn't have blank video every time I connect it. But YMMV.

The dock you use is very important for latency. There is a YouTube video out there that compares a bunch off them.