Newbie with plugging into house question

Thanks for the info and condolenses. Yeah it seems strange that I can't run anything more than lights. Here's an interesting question. I know I need to have the battery turned on for anything to work (I'm assuming this to be the case) because even if I'm plugged in to shore power, it still goes through the battery, right? How do I know that the coach lights are actually being powered by the plug and not just draining the battery?

Also, I'm supposed to have the Inverter turned ON while plugged in so the batteries charge right? I hope that's not a dumb question, probably is. :)