The problem with space elevators is that the cable (or the pipe, as you propose) needs to be able to hold itself together. Imagine taking a 4000 km long elevator cable and holding it up by one end: it would snap under its own weight instantly. That's about what your cable material needs to be able to take. The weight of the car is trivial in comparison.

Does it have to support its entire weight? At some point, won't the pipe have enough centrifugal force that would be supporting it?

The uppermost parts of the pipe (up past GEO) would; that's why it doesn't fall down. However, unless your pipe is strong enough, it'll rip in two, and the part above GEO would fly away and the part below will fall down.