I'm sure that your Question4 is a typo.
For 1>3 your house; coach; R/V batteries; all the same batteries just everyone knows them by a different name , should be charging from all sources. The 12v disconnect should be on to hook the batteries into the system.
Chassis( engine start ) batteries , may not charge from shore power( 110v ) depends on how your coach is wired.
Best way to find out is with a multi-meter, at the batteries with each power source. 13.2v and over, up to 14.3. the batteries are charging.
Rick types faster than I do .