being that you cranked your engine for a while, and everything seemed fine for a while afterwards, leads me to believe that you have a BATTERY issue, probably and battery CHARGING issue. Find if your Converter is actually charging the HOUSE battery(S) while you are plugged in.
As the HOUSE batteries start to lose voltage, many weird and strange things can start to happen, such as the 'flickering' of lights, which is an easy 'sign' that the 12v system, i.e., the HOUSE Battery system is not up to par.
As for your Air Conditioner not cooling, that appliance is a 120v device and will ONLY work when on 120v power, whether a 50amp outlet, a 30amp outlet, or any regular 15a or 20a outlet. If it is not providing cooling, then either the thermostat has lost power and is no longer keeping it in 'cool' mode, even though the fan may still be running, or the cooling is not happening due to a faulty a/c unit. I vote for the former. The thermostat is 12v, so yes, if the HOUSE batteries are getting low on voltage and the lights are flickering, then the thermostat is seeing the same issue as well.