I do not know of a RV that uses the same locking system for the cab and the house doors. That is a automotive issue that would be difficult to surmount. Though I agree it would be a nice feature.
Defiantly go to the Lazy Daze Owners group.
When posting questions about the generator use a cell phone camera to take a pic of the generator face (the green door). That door should come off so you can access the generator for most most of the maintenance work.
The house batteries on a Laze Daze are separate from the cab. The electric door locks belong to the cab's (Ford) electrical system. There should be plenty of room to work on the cab's battery under the hood. If the house battery system is still stock Lazy Daze setup (no changes over the years) the two 6v batteries should be on a drawer that slides out for easier access. Some people remove the drawer so they can add the next size up 6v batteries. My rig was well cared for by the previous owner, but in the short time I owned the rig the drawer's latches started sticking.
Re the box on the underside. I have no clue with out pictures.