Welcome to the club! I can't answer the fridge question, but the furnace will stay on if you are plugged into the tow vehicle AND the furnace switch is not fully off.
Let me explain -- on many thermostats, you can slide the little knob all the way to the left. You think the furnace is off, but it is not. You have to push the knob a little bit to get it all the way to the left. You will feel it click.
Otherwise, if you just slide the knob left without clicking, you may find that while towing in cold weather, you have been burning propane for nothing.
Please don't ask how I discovered this little problem with the thermostat.
Be sure to ask lots of questions!!
2012 Fun Finder X-139 "Boondock Style" (axle-flipped and extra insulation)
2013 Toyota Tacoma Off-Road (semi-beefy tires and components)