Chris
The other specific answers are gonna have to wait a while- don't have the material handy. But I noticed one thing- you seem to be under the impression that 777s have more than one main battery. The ones I work on have a main and APU battery, and unless you want to count the fuel spar valve batteries, I know of no others.
As to the rest, yes the APU battery can only send power to the R/h ASG. The APU battery charger is powered through the F/O instrument bus, and the APU battery supplies lights when towing.
More later