Host Mammoth Question.

Do you just run the pump dry for a few mins or do you have to put some water into the fresh tank first?

I'm having the exact same issue that OP is having, city connection adds a couple gallons to fresh water tank after about 12 hours.

Yes to pump check valve. I’ve had this happen before also. Dirt particle or something caught in check valve allows city water to back fill. Running pump for a while cleaned offending particles and seems to be fine since. Cleaned filter before pump just to be sure. Considered adding check valve after pump but have not as yet done that.

Most likely it is the check valve as the pump will run momentarily even when not being used.

Ditto on checking check valve on the pump.
I did have system that would fill the tank from city water connection on motorhome, but doubt you will find it on camper.

Ok

Could be a stuck check valve. Turn off the street water and run the pump. Alternate a few times. That may get the check valve to seat.

When hooked to city water connection it is slowly filling the water tank. Also water pressure is low like a valve may be lined up wrong.

Most RVs have a ball valve that can stick and prevent water from coming in through the freshwater hookup. A couple of more items 1. Water Pressure reducers can also stick closed. 2. You can always add water directly into your holding tank if you can get the freshwater intake to work and you don't want to fix (like on a trip).

When hooked to city water, it is what fills the water heater and pressurizes the sinks, shower and toilet. I guess an answer to the question is yes...the water pump switch.