1) A vehicle starting at the earths surface has the moment of earths rotation. As it climbs the cable it would need to be accelerated to a suitable orbital momentum. If you don't do this, then it'll "tug" on the cable counter its rotation.
Edit: To clarify why that is a problem. The climber would be accelerated by the tension of the cable, but as that happens, the cable would be bent at the point of the climber, which might be problematic. Also the counterweight that provides the tension would be decelerated, so the whole cable would "tilt". That is all, as long as the climber climbs. When it stops climbing (or is jettisoned), both counterweight and climber would now start a pendulum motion (at different frequencies). That's probably also not good.
2) To reach geostationary orbit at a speed of say 100km/h would take 2 weeks. Traveling 500km/h it would still take 3 days. Unless you jettison the climber at the top, only one climber could be on the cable at any one time, and that would put a lot of strain on recovering the initial cost of building the elevator by putting an upper bound on how often you can send a climber up to anything between every 6 days to a month. I don't think anybody would engage in a project that cost trillions of dollars to get built, and then can only send 12 missions per year into orbit.
Edit: People are pointing out that a second cable would solve this issue. I think it might be challenging to keep the two cables from tangling up across a tens of thousands of kilometers.