Desertryder,
We have done it a lot with an Avion fifth-wheel, and for the same reason. Pay attention to the fuel supply and make sure the generator doesn't run out. You don't want the dogs trapped inside with no a/c.
YZFBOB,
The truck's batteries will be charged from whatever source produces the highest voltage. That may be the truck's alternator or the trailer's DC converter (powered by the trailer's generator). I would guess that sometimes it's one, sometimes the other. When you shut off the engine, the truck's batteries will be charged from the trailer as long as it has AC power from any source and you leave it plugged into the truck.
Aside: If you leave the trailer plugged into the truck without AC power to the trailer, the trailer will drain the truck batteries.
Loren