Yes a 30 amp RV will have limits. If running the air conditioner I recommend setting the water heater and fridge to propane only.
The two small fridges probably are a small draw but start up surge each time it cycles on can be significant.
Don't run your heater and air conditioner at the same time. Probably can't. Furnace runs on 12 volt power and propane. The converter will supply plenty of 12v power when plugged in. Off grid the furnace 12v draw on the battery can be significant.
No reason your 12 volt battery should get low when you are plugged in. Some have a storage switch that must be in the "on" or "in-use" position or you will not get power to the battery for charging.
Yes the converter can draw significant 120 volt power if charging a low battery. At 1 or 2 hours the battery will generally be charged enough that the converter will be operating at minimal power.
The 12 Volt Side of Life