The step extends via the engine battery, or at least on my 2015 model. Yes, the engine will charge the house batteries after the engine has been running for a minute or so. There is already an isolator in the PC. Shore power is better, you don't need a 30 amp outlet as long as you don't try to run the air conditioner or microwave. So a standard 120v plug adapter to a 30 plug works fine.
On my 2015 model, the generator will start via the engine battery but the engine must be running for a minute or two before you get any power to the generator starter.
Make sure, if your house batteries have water in them if they are flooded cell batteries. If they are the original batteries, they may just be plain worn out.
No, do not jump the house batteries with your car. That will only give you a momentary charge unless you want to leave your car idling for several hours.
A standard 12v battery charger will work. Depending on the amperage of the charger, it could take 12 or more hours and that's if your house batteries are in good condition.