You can still have cutoff with the AC running if the car "thinks" it's cool enough. If you sit there long enough, the engine will restart when the car begins to warm back up.
Well it is specific to the 110 plug in effect. If there's enough draw then the alternator must be turning so as not to deplete the battery running through the inverter.
Having the nite light plugged in tells the computer power is needed for what's plugged in. Thus the alternator needs to turn so as not to draw down the battery.