If they live on Jasmine, then they live in Condor Terrace, those are the apartments. I can't remember when Jon got promoted vs when Erin moved to 29. If he was a LCpl when he signed up for housing or if Erin did the move without Jon using POA, then that's where they'd put him and you don't move just because you get promoted (although I'm sure there are cases where it has happened, it's few and far between and usually involves adding family members mores than adding rank.)
The commissary, Housing Office and all housing areas and buildings on that end of base, on base. And you do need to go through the gate (with MP checking ID) to get to them. If you enter or leave while the Condor Gate is closed then you drive to the Main Gate that is open and manned 24 hours.
801 housing is "out in town". However they have added unmanned gates.
The school property is not Base property but is butted up to the base. The school busses do go through the gate to get to the school. I don't know all the logistics of it nor do I even know if only military kids attend? or town kids attend?