Honestly, I'd guess you'd find those things in most of the neighborhoods you list. (I can vouch for everything existing on the North Oakland/R'ridge side, and I imagine they're all there somewhere on the SF side, too.) Where in SF will you be in class, and will it be every day? That could be a big deal if it's not BART accessible or in SOMA (Transbay bus), since it could be a pain to get there. Children's does have a shuttle from Macarthur BART that's open to anyone, I think, so that would be a decent commute from the city if you stick to BART-accessible parts (Mission Dolores, etc.).
Oakland/Berkeley will likely be cheaper and you're more likely to find yards and dog-friendly housing. However, it can be a pain to commute to the farther-out SF neighborhoods. That part of the East Bay is also mellower than many of the neighborhoods you list in SF, if that's a concern. Personally I think the East Bay has better recreational hiking and biking, too (Oakland and Berkeley hills, etc.) but that's debatable.
You might check out SFGate's neighborhood profiles--they give a taste of each of these neighborhoods on both sides of the Bay.
Bay Area Traveler: SF Neighborhoods
and
SF Gate: Bay Area Traveler: East Bay Neighborhoods