Well, actually...
Package your cable up, launch it, move to geosynchronous orbit. Start deploying your cable. Once it gets long enough, say a few kilometres long, tides hold it rigidly pointing towards Earth. You solve the counterweight problem by also extending a cable outwards from your launch vehicle, so keeping the centre of gravity still. Eventually the bottom of the cable reaches ground level and you're done. On the plus side, you now also have a 36,000km long cable extending past geostationary orbit, ideal for interplanetary launches.
> What would we do if several kilometers of ultra-strong cable fell down into the earth?
Not much. If the point at which it's severed is low, the lower part will just fall down and you pick it up and weld it back on. If it's a bit higher, the lower part falls down and burns it. If it's a bit higher still, it goes into orbit.
People have been thinking about the engineering and geometry of space elevators for a long, long time...
