The main items that are NOT powered by the 12 volt
house system are the air conditioning and the microwave oven. Most TV's are also not powered by 12 volts, but some can be.
Bob brings up a good point, often overlooked, that the 12 volt
system for running lights, brake lights, electric brakes, and anything else that's chassis related is a separate 12 volt system.
The converter is a glorified battery charger with enough excess capacity to run your 12 volt items with a dead, or nearly dead, battery.
Your battery (or batteries) is the only connection between the house 12 volt system and the chassis 12 volt system. The battery will charge from the converter when you are plugged into 120 volt AC power. The battery will also charge from the tow vehicle (TV) whenever you are plugged into it. If you stay plugged into the TV and run enough 12 volt items on the trailer, you can deplete both the trailer battery and the TV battery.
The trailer battery must be installed and turned on, if you have a switch, whenever you are towing the trailer. It is the source of power for the emergency trailer brakes via the breakaway switch.