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It absouletely does not clear you through the B, but normally they'll give you your clearance way before you have to ask for it.
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Correct. In fact, be very cautious of using VFR flight following. You're bottom priority so far as ARTCC is concerned. You may be approaching Class B airspace and they won't tell you since they're busy working all the traffic that is more priority to you. Besides, it's the pilot's responsibility to always know where his aircraft is going; during VFR flight following, ATC has no responsibility over that since you're VFR (nor do they want the extra workload). If you're approaching Class B airspace, NO clearance is automatic; unless you hear "Cleared into Class B" from the servicing TRACON, you need to stay clear.
Hence, why VFR flight following is a false sense of security unless one knows exactly what that service provides and doesn't provide.