Yes you can run lights, water pump and furnace off of 12V. But, you will need a generator to keep the battery charged. Even a 1000 watt model will do. Best suggestion would be a pair of 6V golf cart batteries in series. Strongly recommend that you upgrade the Converter to a modern 3 stage type.
If I was on a well I would drain and re-fill my water about once a month. And sanitize twice as often. Our RV is also ready for us 24/7
I know that water being supplied from a municipality that has chlorine added can be safe up to 30 days under ideal conditions. In your case using well water I would research how much chlorine you would need to add for the volume that your tank holds.
I also agree with the poster that suggested not completely filling the tank in case it were to freeze.
Like you and many others my trailer is a big part of my emergency plan.
First. If you have an outdoor shower make sure both valves are off. Second. Check the position of your hot water heater bypass valves. Either of those situations could cause your problem.
Usually there is only one check valve in the system and that is at the inlet connection.
I have the Eureka Northern Breeze shelter. A very sturdy screen house when secured properly. Lots of options when you can utilize the rain curtains as more shelter. And definitely not as fragile as the scissor type "quik me ups".