Furnace and AC

... also true of the electric element in the water heater.  The propane side of the water heater needs a 12 volt source for ignition.  Also keep in mind that the furnace sucks down propane and amps from the battery if you are off grid. 

The fridge will run from 12 volts, propane or AC.  Only use the 12 volts while towing IF you are sure you have a large charge line coming from your TV - if you stop for lunch longer than an hour you'll want to turn it off or switch to propane because it will kill a battery quickly otherwise.  The propane mode still requires 12 volts for the circuit board but it sips propane.  You can go a month on one full tank (if you don't use propane for the stove or furnace).  Personally I've always taken advantage of the 12 volt option on my campers while towing, but with the electronic ignition on the pod's fridge I'm starting to get a little lazy and only use the propane on the road.  No worrying about long stops, etc.

A suggestion for battery longevity, don't run a lead-acid battery to 0% state-of-charge.  Most of the time you want to stay above 50% SOC, occasionally you can go down to 30%, but 0% will kill it after a dozen times.  Use the 50% point most of the time and it will last you for 5-7 years if you also keep it charged.