The only time an inverter will show an AC power is when it is creating it. When plugged in or on generator, the inverter is not used. Its output will go through its own separate transfer switch that feeds certain power outlets. This switch will select either shore/generator power OR inverter power.
I don't know if there is something in your rig that keeps the inverter from powering on. Typically when an inverter has a good source of 12 volts feeding it and it is turned on, it will produce an output which should display on its screen. Even if it is producing 120 volts it wouldn't be feeding the outlets if shore power was available. The inverter will not display 120 volts coming in from shore power, only its created voltage.
Your shore power cord end should not affect the generator supplying power as they are totally separate.
If someone removed the transfer switch and hard wired connections they could have created a very dangerous situation. Maybe the switch failed and they just mickey moused the connection by wiring everything together. This could mean that the generator is receiving power from your shore power connection. Holy crap!
At this point you may want to have a knowledgeable person take a peek at the wiring. It sounds like your 12 volt system may be working OK since you said the solar powered things OK. The 12 volt system is completely separate from the 120 volt system, which feeds the microwave and air conditioners. The TV will probably get power from EITHER shore/generator power OR inverter.
Side note - very old rigs may not have an automatic transfer switch and require you to manually switch something. My old camper required moving the shore power plug from a socket, which was connected to the generator, and then plugging it into the shore power.
Maybe someone with your exact rig can tell us if you are suppose to have an automatic transfer switch or not.