Inverter Power Draw and Power Switch

When the 110V inverter is hooked to the 12V battery and the car is running, I read that the inverter will never draw from the inverter? What I read was in the context of solar with a charger - so similar situation of an inverter hooked to a chargin power bank. I think I know the answer, but looking for a confirmation. Someone once told me current is like water - the stronger flow takes over. So the alternator, having a higher voltage, will always take lead. Is this correct? Or wrong, as in the inverter always draws from the battery and the alternator is charging it.

If I want to switch from the car battery to an aux battery when the engine is off, do I need a an ATS - automatic transfer switch? Can they be programed for specific voltages?

What if I want the charger and the inverter all to switch from 12v to 120V when plugged in, what do I use in that case?