What do you mean when you say "regular" plugin? That means nothing. 15A, 20A? You need a 30A, but I think you know that.
Also, what do you mean "converter attached."
That being said, MANY things use 12V as part of the appliances system. Furnace, WH, refr, AC t-stat, radio, lights, slide motors, jacks.
Your AC would run, but probably trip your shore power's breaker really quick.
Do not trust the monitor panel saying the battery is "2/3" full. That could mean 10-12VDC. If that's the case, your converter that makes 120V into 12V, might not be working.
Also, were you physically lighting the pilot of the WH, or do you just mean you turned on the WH LP switch? That shouldn't have done anything anyway.
The AC blower and microwave are 120V only, so if you have no 12V, that means nothing.
IF you have your battery out, but are hooked up to 120V, your converter should still create 12V. If it's not, you have a loss of power somewhere. Either 120V breakers, 12V circuit breakers, or 12V fuses.