I can answer some of your questions but will need more information to help with others.
An inverter makes 120 vac from 12vdc power source. If it has a built in charger, it will recharge battery bank when outside power (generator or shore power) is on.
If you want to run your entertainment components without the generator or shore power , an inverter will power them from your house battery bank. Depending on the size of the inverter, it may also power the microwave and other 120 volt receptacles.
Inverters can be installed in many different configurations. To find out what yours will power; turn off shore power and generator. Turn on the inverter and check all 120 vac apparatuses for normal operation. Keep notes as to what works and what does not. Usually roof air conditioners, washers and dryers are not feed by inverters.
The controller is in the kitchen only for the ease of wiring from the manufacturer or after market installation.
I would recommend a qualified technician install your upgrade if you do not have a thorough understanding of the wiring of this system.
Mike, RVIA & RVSA Certified Master RV Technician
Amy, Dr. Assistant - Roxie & Mei Ling, four legs each
2000 Gulf Stream Scenic Cruiser 450 hp & 1330# torque
06 Saturn Vue, 06 Chevy Z71 4x4 & 2014 Corvette Z51 M7