Forget all this space elevator malarkey, just put a winch on the ISS and lower a line down with a hook, snag stuff as you fly past and then simply hoist it up. As long as the combined weight is less than the ISS it won't pull it out of orbit (you could add a couple of retro's just to be on the safe side).
What you propose is called a sky hook. Like others said, it won't work the way you described it.
Snagging something on a hook travelling at 17,000 mph is not easy. Lowering the hook in the first place would also be tricky - it would burn up in the atmosphere.
Well these days there are materials that are strong and light enough that such a cable could exist - the same reason for the increased talk about space elevators. However to solve issue of a really long cable whipping about on earth, ISS would need to be put into a geostationary orbit from its Low Earth Orbit...
A 300 mile long cable strong enough to lift anything would already weigh more than the station, never mind adding anything to hoist. Also, a cable and hook whizzing along the ground at a few hundred miles an hour is not a smart idea for obvious reasons.
Yes you would pull it out of orbit. Each time you lifted something, it would lower the station. As mentioned the cable alone would weight more than the station. And no cable is strong enough anyway. And the end of the cable would travel as the station went around the earth (although at a different speed than the station itself, a whole other problem) Etc etc etc. Not gonna work
