1. The onboard tank should supply all of your water needs, including the toilet.
2. Personal choice. I would shut off the pump when you're not in the trailer/coach. But then, I leave the coach when it's hooked up to the city water so I'm not sure that's even necessary.
3. Correct. Turn off the water pump when hooked up to city water.
4. It depends on what type of fridge you have. If it's 120 VAC and propane, the propane burner should start automatically when you lose AC if it's set to Auto. Looked at you fridge control panel. Unless it has a DC setting, then it won't run directly off the battery. Depending on how they wired it, you may be able to run the fridge on AC off the inverter, off the house batteries which will charge when driving. Consult you coach's manual to see how it's wired.
Don't worry. It'll all make sense in a few years :)