If you're going to keep your trailer plugged into your house outlet when not using it will help keep your batteries topped off. If you're not, then I'd strongly encourage you to either disconnect the batteries or install some sort of kill switch. There are several things that will cause the batteries to discharge even though you swear that you didn't leave anything on. Coming out to dead batteries the day before a trip is not a fun experience.
And leaving your rig plugged in to your TV while you're having lunch isn't going to drain your car battery. I'd be inclined to unplug only when your ready to stop for the night.
Have a wonderful time with your new rig! And when you can share a few pictures with us. We'd love to see your dogs & how they're enjoying their new moving home.
On the road and happy with Jed, my four-legged camping buddy.
'06 Eddie Bauer Expedition
'13 Starcraft 265RLB Autumn Ridge
You shouldn't run your refrigerator for long periods of time when you have it parked unless the trailer is level. You can run it while towing but not parked out of level for any length of time. The RV refrigerators work by boiling Ammonia unlike the home type refrigerators that run on freon. So they need to be level to keep the chemicals from crystallizing when heated unevenly.
As far as turning the refrigerator off when gassing up at a gas station you'll find that most people don't turn them off including me.
2010 Rockwood Signature Ultra Lite 8315BSS
2003 Chevy Silverado 1/2 Ton Extended Cab
As for the propane at gas stations. Here's my routine.
I pull into the station but never pull the trailer under the awning or near a pump. It's the first pump. Before pumping fuel I go into the trailer and turn off the fridge.
I pump the fuel and leave. In the clear area (truck stops are best), I stop and turn the fridge back on then hit the road. It's the igniting I fear at the pump.
You are going to get a bunch of different answers on this one.
For me, even when I had a gasser, I set the fridge to auto and left the gas turned on. Never turned it off when pumping. I figure that the flame for the fridge is many feet behind and above the fill on the truck. IMHO the vapors should be well dispersed before they get anywhere near the fridge vent.
I do not turn off the fridge or the LP until I get back home and unload and shut off the rig. That could be a couple of weeks afer I turned it on.
One benefit of leaving it on auto and not turning it off is that I have never been surprised of having warm food upon arrival from forgetting to turn it back on after fueling. In 100+ temps even the best fridge will warm up some during travel.