The plug or receptacle shorted out. Not enough current draw during the "arc welding" show inside the receptacle box to trip the shore power breaker you're plugged into.
Replace plug/receptacle and you should be fine. I'd put just a new plug on the shore power cord and new receptacle if you have enough un burnt cord on both ends. Note whatever appears melted is probably damaged another foot? back into the cord or wire insulation. May have to chop a couple feet off the cord. May have to just replace the wire from the receptacle to the campers breaker panel.
Good thing you caught it
Yes just charge batteries as if they were a single 12V and no problem using power while charging.