Most of the lights are likely 12V and if your converter isn't working properly you'll run down the battery and the lights won't be very bright, and if the battery gets low enough you'll have no lights. The thermostat is also most likely 12V which would explain why your AC quits when you use the lights.
As was recommended check the voltage at your battery. My guess is that your converter isn't putting out enough voltage to charge your battery well enough. Your battery could be toast by now, and if you don't act quickly it will definitely be toast. Depending on your setup you could get by with a new converter but you'll have to check to see what you have.
I discovered all this first hand just a couple days ago. Post back when you have checked your voltage at the battery.
