Yes, the cables could just terminate in space. The only important thing is for the center of mass to be at the point of geostationary orbit. If you use 70k km of cable instead of 35k (half below GEO and half above) then the center of mass would still be at GEO and the system would still be stable. The advantage of this is that the end of the cable would be traveling at far above orbital velocity, so you could launch things directly out into the solar system without having to burn rocket fuel to move from a geostationary orbit to an earth-escape orbit. The disadvantage is that you have to make twice as much cable.
Well, it depends on whether by "fantasy technology" you mean "technology that we have a good idea how to build and is probably a few years or decades away" or if you want it to mean "technology that we have absolutely no clue how to build and probably won't develop for the foreseeable future."
The minimum strength for a space elevator cable is around 65 GPa, but most designs call for a cable material with a strength of around 120-130 GPa so that there's a comfortable safety factor. We have already developed carbon nanotubes with measured tensile strengths of 63 GPa, and calculations indicate that "ideal" nanotube ropes should have tensile strengths around 300 GPa. So we know of a material that's strong enough, and we are getting progressively better and better at making it. Of course it's not clear exactly when 100+ GPa nanotube cables that are appropriate for a space elevator will be developed, but it's definitely a technology that everyone expects within a decade or two - we aren't talking about wacky fantasy technology that we have no idea how to build.
A cable that would let you pull the entire assembly down from orbit without breaking would have to be a LOT stronger than a cable that merely hung under its own weight.
The cable will have a density of about 1.3 g/cm^3, and is less than 1 cm thick along most of its length, so no one is ever likely to be hit with 100 kg of it. A better analogy would be having two airplanes flying over Tokyo with a long, thin piece of nylon rope strung out between them. Then they cut the rope loose. The rope isn't going to crush anyone or damage any buildings when it lands, it's just going to drape itself over the city and annoy people.
On a side note, Tokyo would never be hit with the cable anyway because the cable has to be anchored along the equator. Sorry, no space elevator for Japan.
You are forgetting that entire cable is already rotating exactly in sync with the surface of the earth when it is cut. Yes, of course eventually coriolis forces will cause it to "wrap around" the earth, but the cut would have to be extremely high (thousands of km up) before they had any significant effect. Of course the cable isn't going to coil itself into a nice little pile right at the base of the elevator, it's going to be subject to wind etc. as it floats down to earth. The point is that there isn't going to be any significant "wrapping" of the cable around the earth unless the cut is made very high.
Edit: And if you're still really worried about it, you can anchor the cable on a ship over the ocean or something.
As for problems that don't break the cable, one of the nice things about the space elevator is that it is self-stabilizing because of the earth's rotation and the centripetal force being exerted on the portion of the elevator system that's beyond geostationary orbit. If you simply pull on the cable for a while it will deform the cable slightly and pull the whole thing down a little, but as soon as you release the pressure it will begin to drift back to its original position. This is pretty important, because otherwise the whole thing would eventually become destabilized by the changes in angular momentum that occur when cars move up or down it.
NGM, you seem very hostile to the idea of a space elevator. It's perfectly fine that you're skeptical, and there are indeed many major engineering challenges associated with building one. But you might want to remember that a lot of serious physicists and engineers have evaluated the idea and they generally agree that it will be feasible relatively soon. It's not like this is some wild idea that a crackpot came up with on this message board; space elevators have been seriously studied since the 1970s. You've only been hearing about them in the popular press recently because it has only been in the last 10 years or so that we've gotten close to having the technology to actually make a cable that is strong enough. Of course that doesn't mean that we will necessarily ever be able to build one, but most of the things that immediately come to mind as potential problems have already been very thoroughly evaluated by professionals.
