The solar should easily maintain the battery bank during storage easily, assuming you store it somewhere with at least a little sunlight.
You generally can run the generator while en route, and the air conditioner. People in motorhomes do that quite frequently if the dash air conditioning is insufficient to keep the interior comfortable. Difficulties may come if you're talking about e.g. a portable generator in the truck bed and a wire dangling between the trailer and the generator from the actual connections and so forth. I guess you're probably talking about a permanently installed generator, in which case there should be no trouble at all.
The tow vehicle should now provide some charging power, but it would not be a bad idea to verify that. You can't generally expect to see a very large charge current. It's probably best to think of it as enough to keep up with the parasitic loads and the fridge control board than as a way to recharge a depleted battery bank, doubly so if you're not driving for many hours.
An inverter to run the TV and DVD player would be very simple to hook up; you don't need a particularly big one for that. You could probably plug it into the 12V lighter socket at the television if your trailer happens to have one there, and not have to do any permanent wiring.
Powering the microwave from an inverter is certainly possible, but to do it effectively you'd need a larger battery bank and a decent plan to charge it afterwards. Four batteries is probably about the minimum to reasonably use a typical microwave, and even then extensive cooking is asking a lot. It's often more effective to start up the generator for that if you are able to do so (or cook on the gas cooktop).