You can also solve the problem of the downward pull by having the satellite at normal geostationary altitude, but have the tether extend well beyond that. The tether below pulls downward on the satellite, but the tether above pulls upward on the satellite. You can get those to balance out, and that actually works out nicely for an elevator. You can send things you want to launch all the way to the far end, and then let them go, and they've already got a very large velocity boost.
Of course, the real problem is that to actually build one of these, you need a material with a tensile strength to mass ratio a few orders of magnitude higher than anything we know how to make. Carbon nanotubes show promise, and have a theoretical tensile strength high enough, but currently, we can only manufacture them a few millimeters long (and the tensile strength of a bundle of separate fibers is significantly lower than the strength of each fiber individually).
Conserve energy. Commute with the Hamiltonian.