I'm going to be brave and throw out some suggestions here. If we were on one of the Old People's site, I would get thrashed at my answers. However, they should be good enough for what we do ;)
The Travel/Park switch is probably what you guessed, except that it probably uses the same circuit. Probably the coach battery, but the switch will prevent a drain. I helped Cat and Bryan with theirs, and I think we just wired it hot all the time. Just turn it off if you aren't using it!
One way to tell what battery something is running off of is to kill the house batteries. You should have a disconnect for that. Mine is in the battery compartment on this coach. Kill the house batteries and see what is still running.
Is the house battery charged from the chassis (engine) electricals while driving?
Sometimes. Mine do on this coach, but others did not. If you have an inverter, you can watch that panel while the engine is running to see if there is any charge going on. I would be that there is.
Is the house battery charged when we're plugged into a 30 amp spot?
Yes. This is what the converter/charger does. Hopefully it is a 3-stage charger where it does bulk, then trickle and float charges. It should maintain a charge while plugged in.
Is the chassis battery charged when we're plugged into a 30 amp spot?
I would have to say probably not for this one. Ours is not on ours. You will probably have a "Battery Boost" switch. This is used when your chassis battery is drained after sitting for a while. It "jumps" your chassis with your house batteries.
Do they both share a common ground, or is there a separate ground for each?
Yes, they share the chassis as the ground. For the 12V, you can count on that chassis ground every time :)
Hope this helps. The 12V is a bit of a mystery, but figure it out and never be out of power again!!! :)
--kev