Yes, I think the Main switches need to be on in order for the inverter's charger to keep the batteries up. Only the solar panel can charge them with the switches off, and your Quonset shelter takes solar out of the picture. My banks stay charged year-round with the switches off, but it's because the solar panel is exposed all the time, and easily compensates for parasitic losses to the engine and tranny control modules that drag on the chassis set even with the switch off.
As a new owner, some insecurity is expected until systems familiarity is down. As long as you've made sure the really big electron hogs are off, you shouldn't fret too much. You could always unplug everything to be sure, or flip all the breakers off except the inverter. But if you rely on the electric side of the hydronic unit, that's risky, as can be a portable electric heater although I've used them during cold snaps in the past. Neither should be necessary if the coach has been winterized with RV antifreeze.
Don't use too long of an extension cord, if any is necessary. Use the shortest length of the big 12 gauge size as possible. The last thing you want is a 100 foot 16 gauge cord, where it loses a ton of power to voltage drop. And don't skimp on the adapter between 30/50 to your 20 amp source; inexpensive ones can overheat when taxed, melt, and catch fire before the building's breakers kick off - been there, done that. Get a nice heavy duty pigtail-style adapter or adapters.
As to the auxilliary compressor, it shouldn't come on unless you set the air leveling to automatic leveling. Regardless, these coaches leak air over time, and if one corner gets very far out of whack, you risk a body twist that can crack your windshield - also been there, done that. I don't park on my pad and leave it stored in Travel Mode anymore, nor do I try to level it. I've made my "pad" relatively flat side-to-side, although it's sloped front-to-back. All I do now is dump air so the whole coach goes down evenly. At or near bottom, I stop.
Because it is a simple front-to-back tilt and not side-to-side, and no air leak can throw things very far askew, no harm can come. Plus the wife can easily negotiate the first step. The tilt is not enough to be detrimental to the fridge when we next power it up prior to a trip, but I usually level the coach then anyway - 12 hours or so doesn't provide any leak time enough to do harm, and auto-leveling activates the auxilliary compressor to fight that anyway.
Joel