Also have a 2012 Greyhawk. As stated above, the inverter only provides 120 to the outlet behind the TV. Unless you have disabled the relay, the outlet will lose power as soon as you start the engine to your Greyhawk. It's a safety feature so the driver isn't watching TV going down the road. It can be disabled, and there are a number of threads on this forum that discuss how. As far as your other outlets, the generator needs to be running for those to be energized. The fridge can run off of 12volt and propane, so the generator does not need to be on to keep things cool. I always leave the propane on. Never had an issue.
Brian, Mary, kids, beagle - Moose, and mutt - Maggie.
2012 Greyhawk 31FS
2013 Ford Explorer Toad