Firstly, rain water isn’t really for drinking, and you haven’t got much of a roof to collect it. Its not practical to heat anything by battery power, not with the amount of solar panels you could fit on the roof anyway.
3g dongle, yes.
LED lights, yes. batteries can be kept topped up by panels but most of your charging is normally done with a split charge relay and the alternator.
Get a diesel. You can get a vapour take-off LPG tank for air heating, water heating and running the fridge. If you got a petrol, you would still need separate LPG tanks as one needs to be liquid take-off and one vapour. I have a tank, saves valuable space over cylinders and I can top up as necessary instead of waiting to empty a cylinder.
Wood burning stove, technically yes but insurers will run a mile, they take to loads of room, and make you look like a traveller!