When you are completely hooked, it wont stand still ,long enough to worry about any of this advice as you will be away in it all the time!
Propane, (red bottles) is you best bet for gas as it keeps working when the weather gets very cold, you can leave it conected just turn the tap off on the top of the bottle,
As for you battery, i would trickle charge it, or just plug thre caravan into a scoket and leave the zig unit charging it for a while, but I think it would be bette of in the carvan as it would be out of the way of the frost etc, and cant get left behind!
Though if vandals, or security is a risk, of course take the battery and gass bottles out! We keep a caravan at home and leave the mains plugged in all the time, then i flock the charger on from time to time, and use a littel £20 dehumidifier to keep it dry and fresh.
I dont think the tyres mind much standing now they are modern radials but it can hurt to cover them, I never do but mine, but they dont stand for long, though it will be worth checking that they are all the right tyres for the caravan, not just car tyres (unless a very small light caravan) and that you have a spare etc.
SO what make and model have you bought then?