I owned the "push-pull" 80-200, updated that to the 80-200 ED, then eventually purchased a 70-200 off a friend who was downsizing to mirrorless. So I've had all three.
The 80-200 has a few disadvantages to the 70-200 VR, but it is a perfectly fine lens to use on a modern camera. It lacks some modern features like VR and AF-S, so maybe a little outdated, but not obsolete.
The main obvious difference is VR. Not helpful shooting sport, but if you're shooting bird or other wildlife at 200mm, very useful.
The 70-200 also couples very well with the Nikon teleconverters. They are expensive but you lose very little sharpness with a Nikon TC and the 70-200. Again useful for birding, safari etc
Both lenses are very sharp. Optically the 70-200 has more ED elements, and more elements overall. I'm sure it's improved over the 80-200, but I could not detect any difference in sharpness between the two (but never compared them side-by-side).
As for focusing, I never had problems with the 80-200. I used it for shooting fast moving sports, on a D90, then a D700. The 70-200, being an AF-S lens, will focus a bit faster and more quietly. So quietly that sometimes I wonder if it's focusing at all, whereas the 80-200 would snap into focus, and you could feel the torque on the lens.
I don't know about focusing issues with the 80-200 on the D7000. There are always people with either a bad copy of a lens (or blaming bad technique on equipment). I can't imagine there's a real and common issue with the combination. But if you have the opportunity to test the lens on your camera, or are able to return it if you're disappointed, that would be good.
I only bought the 70-200 because I had the opportunity to buy it from a friend at a good price. I don't regret it, it's certainly an improvement in some ways, but I for my (hobbyist) uses - I wouldn't say it justifies the price difference over the 80-200 (although the 70-200 is worth every penny if you're using it professionally or can otherwise afford it)