There are two batteries per plane, is that right?
Yes
One for the APU and one for the rest of the plane's electrical requirements?
Yes
The various battery problems encountered in the 787 involve both the APU battery and the main ship battery, yes?
Yes
Are these both Li-ion and made by Yuasa?
Cells (Li-Ion) are made bei Yuasa, Batteries are labled "THALES"
And are they both physically in the same electronics bay?
No, main batt in fwd E/E, APU batt in aft E/E compartement.