Your air conditioners, microwave, TV, and all of the outlets operate on 120V-alternating current power that must come from either the shore power plug or the generator.
The overhead lights, the furnace, and water pump all use 12V direct current from either the battery or from a converter that takes power from 120V and then converts that into 12V-dc to recharge the batteries and supply all other 12V loads.
The RV refrigerator can use either 120V power or propane, but it must have 12V power to operate the controls. The furnace and water heater also use propane but require 12V power for their controls.
If you have not overhead lights when you do not have 120V supplied, most likely the coach batteries are bad. You should have 1 battery to supply the starter for the chassis engine and a different one for the starter of the generator and the 12V lights and appliances. If the generator will start, that should mean that the coach battery(s) is/are good as they supply the starter. It may be there there is an isolator switch for the coach battery that is turned off to save the battery and so disables the lights, but can still supply the starter.