If you have the temp inside at 72 (F) when you start towing, unless it's extremely cold, you won't have to worry too much about interior temps dropping below freezing during the day. Check at lunch. If interior temps drop close to freezing, you might run the furnace to warm up the trailer while eating lunch.
If it's extremely cold, before towing you can use compressed air to blow out the water lines, faucets and toilet to keep them from freezing.
You can run the furnace while towing, but just like running the refrigerator, it adds some risk. At fuel stops, always assume the guy on the other side of the pump is going to spill some gasoline (even if you're pumping less explosive diesel). Gasoline vapor and open flame --> BOOM!