Yes your battery will charge when it is hooked to shore power, and As you have a F150. It will NOT charge your TT battery UNLESS you install the relays into the fuse box. On a 2013 there are 2. Your brake controller will work with out them. But the truck will not charge the TT until you install them. FORD does NOT install them but does supply them. I found mine in the dash pocket. VERY important that you install them.
Q2. Yes your Fridge mother board is run b 12 volt as well as your water pump, and Water heater mother board.
Q3. I unhook my battery and hook a solar charger to it when not in use. I leave it on the TT.
