Furnace on while moving

Of course. No reason not to. It's done all the time.

Run the furnace, air conditioner(s) refrigerator and water heater while moving if you want.

Bill

go right ahead. just remember to stop short of gas stations and turn off the furnace and the fridge before pulling in to fuel then remember to turn them back on after you leave the pump area

I guess it's possible, but I wouldn't. Considering fridges blow out, I would expect the furnace to blow out.

Been running with fridge and heat on for years. Never a problem. Fridge is run on 12 volt and heat is ignited by a glow plug. If I may suggest putting all of your bedding in the overhead stuffed up against the front wall to use to better insulate. And definitely use the overhead vent insulation of your choice.
I have never run the temp above the lowest setting of 50 degrees.

I have pulled my TT many thousands of miles and I have never had the fridge blow out. My furnace has never blown out in 60 mph winds so why would it while driving? The thermocouple would shut the gas off if the furnace stopped working properly.

Yes Yes and Yes
I do it all the time with no blowouts
Just remember to turn off the propane while fueling up.

I run mine, but not at the "lowest" setting.
I set it to about 60 degrees or so.

About an hour before I reach my destination, I turn it up warmer with the bluetooth t-stat I installed a few years ago. THe vieo shows it in a travel trailer I had for a short time. I moved the t-stat to my AF992.

When traveling in sub-freezing temperature, the heat stays on low to prevent frozen pipes and doggies. No problems yet.

I made sure I bought a truck with a back window that opened. I use a "donut" between the camper and truck and there is a nice flow of air into the camper if I leave a vent partially open. Whether is it hot or cold, the air from the truck a/c or heater will keep the camper at a reasonable temperature.