How do Class B transitions work in the US?
Is requesting VFR flight following in a class bravo the same thing as requesting a class bravo clearance?


No. Flight following just means that the aircraft is receiving traffic advisories. A VFR aircraft needs explicit clearance to enter the Class B. The pilot should net ATC know that they intend to enter the Class B.


Does requesting a class bravo airspace transition automatically include VFR flight following (radar traffic information service)?


Yes, ATC must separate VFR aircraft from VFR/IFR aircraft while in the Class B. (7110.65X Section 7-9-4)


If I am cleared into the bravo airspace and the controller gives me a heading and altitude to fly that clips the airspace of an underlying class C / D airspace, have I satisfied the requirements to enter said airspace (class C - 2-way communication and squawking mode C, class D - 2-way communication)?


If ATC has given specific instructions that take a pilot through other airspace, then ATC should coordinate this with the appropriate controllers. Per 7110.65X Section 7-9-2 the controller should also either vector aircraft to stay in class B or to inform them if they are leaving it. Pilots should not change frequencies on their own unless asked, and the pilot should ask the controller if they are unsure.

Also, note that:


  Assignment of radar headings, routes, or altitudes is based on the provision that a pilot operating in accordance with VFR is expected to advise ATC if compliance will cause violation of any part of the CFR.



Why does ATC provide traffic advisories for VFR aircraft in class bravo airspace at all if they are on the hook for providing separation?


As described above, ATC is required to separate VFR aircraft from other aircraft in the Class B. ATC is instructed by 7110.65X Section 2-1-21 to:


  ...issue traffic advisories to all aircraft (IFR or VFR) on your frequency when, in your judgment, their proximity may diminish to less than the applicable separation minima.


Unless a VFR aircraft has explicit instructions, ATC does not know for sure what they will do. And IFR aircraft may still need to perform unexpected maneuvers like a missed approach. Traffic callouts make it easier to apply visual separation if needed.