No, it's physically impossible. Even the most advanced material possible couldn't hold the strain that would be required for Earth. Technically it's right on the limit, but that's ignoring that we have an atmosphere that's going to exert forces on it too. On Luna it might make sense.
The top can be anywhere. It would just require adding force to it at/after release. That's trivial. We already know how to make rockets, or it could be something that pushes or throws it. Compared to building a space elevator, speeding something up is easy.