First question: Yes, looks like the heat for the tanks
Second question: You can eliminate it but if you are in a campground with high water pressure you could potentially have some overpressure issues. I will admit to removing mine in both my Argosy's. You can add an external regulator if you get into a high pressure situation.
Third question: Some '60's AS had a valve to fill the fresh water tank from the city water hookup, but by the 70's they were eliminated. Two reasons: people tended to leave them on and overfill and damage the fresh water tank, and there is a potential cross contamination of the water tank becoming connected to the city water input which is a code violation.