Yes, every motorhome I have ever heard of charges both chassis and house battery banks from the alternator (assuming the systems is functioning as designed).
And, yes your alternator is large enough to handle the load of running your residential refrigerator from inverter power while driving. The only qualification I would put on it, were it my equipment, is that I would NOT do this if the battery banks were deeply discharged. Said another way, I would not load up an alternator charging a deeply discharged large battery bank AND powering the refrigerator through the inverter.