Hi Mbearden.
1) You're the second person in the past couple of days with problem number 1. By far the most likely is that this is a leaky check valve at the water pump allowing the city water to backfill via the pump into the fresh water tank. I suggest you first disconnect the city water and second pump some water out of the fresh water tankto prevent any leaks from overflowing that system. You can use you water system just fine by refilling the fresh water tank as needed via a hose and using the 12V pump to provide pressure. In fact, that is the only way I ever use mine.
After getting the overfill situation under control the next thing would be to confirm the check valve is not closing. It is located at the water pump. You can test it yourself by disconnecting the suction line from the pump to the tank and then reconnecting the city water. If water runs back out of the pump suction line you've confirmed the leaky check valve.
Or, since this is a new trailer, just take it back to the dealer and tell them to fix it

2) Nope, its a 120V TV. Silly isn't it? Rather than waste energy on a 12V to 120Vac inverter I just sold my TV on Craigslist and bought a dual 12V/120V TV on Amazon for $130. Bolted right up. You'll need to install a cigarette lighter socket near the TV to run it. That can be connected to 12V at the feed to your audio system. You'll need a voltmeter but its otherwise a very easy mod.
3) On your tank measuring system, if there's no lights at all when you push the buttons and you have 12V available elsewhere then I'd take that back for warranty repair too.
This is all typical new RV stuff, stick with it and it will soon be sorted.