While we are on the road can I leave the house battery on. Also if we're parked in a rest area or Flying J stop should we leave the battery on if we're not running the generator.
Assuming house batteries are in good shape and have 100% charge after running down the highway you should have plenty of hours of use before needing to charge them again.
Not knowing what your coach is or what your battery bank is it is hard to know whether you have 15 or 24 hours of battery life before getting to 50% depleted.
General answer is you never need to disconnect the house batteries until you put the coach in storage. I never turn mine off unless I am storing without a connection.
If I'm understanding the question, the purpose of running the generator is to generate 110/120 volt power. It's not related to the "house bank", the batteries that run the 12-volt systems, unless you're using it to charge the house bank, but that's another issue.
Yes and yes. You have to have 12v to operate the fridge, if it's an RV fridge. You shouldn't have to turn the batteries off for a short time, unless you have a draw on the batteries somewhere, but that's another issue.