I know the chassis batteries should be charging, but I'm not sure about the house batteries. I also know the furnace should run off the alternator when in motion, but I don't think the AC units will--I believe you have to invert or have the generator running to run AC.
This winter we did a cross country trip and after each day, my house batteries were way down (after running the furnace while underway). Turned out I had a bad relay which prevented power from the alternator to run the furnace.
So I'd be interested to know if the alternator is responsible for charging the house batteries. I think someone told me that if the chassis batteries are charged, power will be routed to the house batteries in that priority, but I'm only telling you what my bad memory thinks it read.
Looking forward to hearing more from more learned sources!
Just read the first reply on this post...fills in some of the gaps.