Have you checked the output of the converter?
When you drive the engine charges your house battery. Parked/hooked up to shore power the converter charges your house battery.
Your interior lights, thermo stat on furnace and fridge ALL need the DC side of your coach. They don't run off the AC side of your coach.
You have 2 separate problems. An AC side problem and a DC side problem.
Check the GFI's for the AC problems even if the button is not tripped the GFI's often go bad and need to be replaced. Did you have multiple power items plugged in? Coffee pots, crock pots, hair dryers etc can NOT be plugged in when the air conditioning is on. They trip the GFI's. And make sure your shore power plug is snuggly connected and no corrosion on the plugs.
Check the output of the converter for the DC side problem.
