California, not the US in general.

Depends on the state and the public/private designation of the university. There is also the (mostly unanswered in the courts) question of whether private university graduate students are primarily employees or students. Read up on the kerfuffles at NYU and Columbia University to see how gnarly this all gets.

It's worth noting that graduate students were not able to unionize until recently at many private universities in the US. Even though they now can legally unionize, many remain unionless for one reason or another.