frsh water tank

Very common in all Types of Rv's. Check valve leaking. Take pump apart and clean. Rebuilt kits available on Amazon. Install a check valve between the pump and the FW tank. 3 fixes is all I got.

Never seen that in a travel trailer but it's fairly common in MH's.

Some RVs have a valve to fill the tank, if it is open or leaks, the tank will fill. If the check valve in the water pump leaks, that too will fill the tank. Common problems.

The city water should connect to your water lines after the 12v DC pump, which has a one-way valve so water can only flow out of the tank/ out of the pump.

You would need to add a ball valve off a Tee after the pump and route it back to the water tank, or the water line before the pump to flow water from city to tank.
You would need to do this manually and monitor it to turn it off when full as it would run constantly into the tank and out of the air vent or fill spout.

No it should not be able to reverse through the water pump and fill your fresh water tank.
Have not experienced that in 4 trailers.

Not in any trailer I have owned or even heard of.

When you are hooked to city water all those little plastic water lines are pressurized to whatever the park is supplying you or what your regulator allows. Safer to fill the tank and use the pump.