Disconnect starting and house battery

While on Shore power does it keep the engine battery charged/topped off, does it have the battery control center, ?
I would unplug shore power first while doing any battery work

No need to disconnect house batteries except for cleaning and spraying the terminals on those batteries, engine battery and house are only connected when charging from alternator or shore power converter depending on what charge control connection device the builder installed
You do not need to be unplugged because of starting the engine,
You should be unplugged from shore power when working on the battery cables, house batteries will be connected to live converter, positive cable for house battery will be live when connected to shore power,
Any accidental touch to vehicle metal would be a short circuit and damage the converter
This will also shut OFF any cross over engine battery charging from the converter via the BBC/battery control center IF you have one, many MHs have the BBC, might be more prevalent in class A. Than in C or B