Green should mean that the battery is connected to the trailer system. Red simply means that the battery is disconnected from the trailer system and the components won't drain the battery while sitting. It has no direct meaning to shore power connections.
Where you leave it while in the driveway depends. Do you have it plugged in or not? If so, I would keep the switch on.
I've never had the trailer plugged in with the switch off, so I'm not sure how it would behave. But I don't think it would help you. The converter should recognize when the battery is fully charged and shut off, so it shouldn't hurt to leave it on.
Unless you are using the trailer (or leave the refrigerator on) in the driveway, it really doesn't benefit you to leave the trailer plugged in. That just leaves the trailer more susceptible to electrical damage and could shorten the life of the battery since it would never drain much at all. If not used, I would just unplug it and shut off the battery switch. Plug it in/turn the switch on when you turn the refrigerator on to get ready to go.
