Safety has many sides. As far as fueling, the basic rule and law there is to be NO open flame near the fuel pumps. What that means as far as our trailers are concerned the propane
needs to be turned off when fueling.
Now as for running fridge
while traveling. I've done it a few times but never trusted the tow to keep the house battery
charge at the same time. I even have 150 amp alternator in my Dodge Dakota with full tow package. From experience the stuff in the fridge
doesn't get warm with the fridge
not running for a couple days in 90°+ high temps. If you're worried about it a couple of frozen water bottles put in the fridge before you leave home will do the trick and you'll have battery for lights
when you get to your camp destination.