chucky
Line Up and Wait
1- OAK center is not the controlling ATC for SFO Bravo, so no they cannot clear you into the Bravo. NorCal can because it is their airspace. SFO tower does whenever you take off. However, if you call them to enter the Bravo from outside, they will likely tell you to call NorCal.
2- Ask OAK Center for clearance into the Bravo or Charlie, that'll nicely remind them thay you're there. They'll likely tell you to remain clear of the Bravo/Charlie and then hand you to approach, who will hopefully let you enter.
3- If you enter the SFO Bravo w/o clearance from NorCal approach you are in violation. If you enter the OAK Charlie w/o radio contact with NorCal approach you are inviolation. (Notice: Clearance for Bravo, Radio contact for Charlie)
1 - I'm not sure about Norcal and SFO, but I believe in New York's airspace, helicopters wanting to transition a class-B surface area just call up the tower.
3 - You can also be talking to OAK tower, which is who Hayward will hand you off to (or at least, that's been my limited experience here).
I don't know exactly how things work as far as airspace ownership goes, but I don't think Norcal owns all of the class-B or -C airspace around here. When I was taking a tour of New York TRACON, one of the controlers was explaining the issues associated with flying around LGA, and anything that took you through the LGA surface area, up to 2000MSL was entirely up to the tower (although you'd likely be talking to New York approach, and they'd coordinate for you). It sounded to me like LGA owns that part of the class-B airspace, and I would guess a similar situation exists here.