Executive Summary: I really like the prime over the zooms.

I've shot Hapkido (very similar to TKD) using an 85mm f/1.8 prime and with a 60 mm f/2.8 prime on a D200 and D70-- the D90 has better lowlight capabilities than those two cameras.

Here's the 85mm on the D200 at f/2.5:

This might be a bit of a cheat, however, since there was light from the outside and bright fluorescent lights. Your kids' do jang might not be so well lit.

And for basketball for my niece, I went with the 50mm f/1.4 (used for $100). The lighting in that gym was terrible. I shot ~150 shots, and only one was close to being a keeper, the lighting was so bad and they just moved too quickly to get anything. I haven't shot figure skating.

1. I have many sparring shots where 1/250th of a second just isn't fast enough. When the impact comes, it comes quickly, and 1/500th or faster is necessary to freeze the action. This one's at 1/250th and similar lighting conditions in the same room:
2. A typical test lasted (for us) between 10 to 45 minutes for each belt level. The 70-200mm f/2.8 zoom lens is very heavy; if you want to be handholding that thing for a long time, go for it. A monopod might be better, tripod might be too cumbersome, but check with the instructors about what's permitted. I like to get different angles, and that means moving faster than a tripod will allow. Large pro zoom = big weight, fast prime = less weight.
3. VR gets you nothing in these situations. They are moving quickly, and the extra stop or two (or three or four or whatever) doesn't matter, because those stops are for the lens not moving, not the subject. So if you decide to go with the telephoto, go for the non-VR version (unless you see yourself using the lens for other purposes).
4. That softness when wide open is up to some levels of interpretation, I think. Having said that, I really dig on the 85mm's f/1.8's look at f/2 to f/2.8. But the 70-200mm at f/2.8 can be just gorgeous.
5. Focus is tricky when the focus plane is that thin, and will take practice. It took me a few years to be able to get (what I think are) decent shots of martial arts.
6. Cropping can be tricky. It depends on how far away you are and what kind of shots you want. While I really like the 85mm's look, the 50mm f/1.8 or the 35mm f/1.8 might be better suited for the closeness you have to the action. I'd recommend first using whatever lenses you have and starting to experiment with composition (regardless of whether or not the shots are tack sharp) to determine which focal length you want to start with.
   Good luck!