This is a bit of a fast induction to shooting wih a DSLR, where you're not getting the results you're anticipating and potentially getting upset, thinking it's the camera. Why didn't you start with your underactive family pet, flowers or children like most people!!!!
It's not you or the camera, just the situation
1 - You're shooting at night, it's dark, so autofocus will be slow, especially with a 'slow' lens
2 - You're shooting something at night that's moving. Even if you're lucky with your focus, you'll possibly end up with motion blur due to the slow shutter speed
3- Even if you've got a fast enough shutter speed, you'll have high ISO noise in your image because it's so dark
This is where fast lenses come in, in this situation. You're shooting at 1/60th second at f5.6, ISO 5000. Now the D7000 should be capable of producing a respectable result if something's stationary with those settings, but the 1/60th second is too slow. 1/250th second is more like it for moving dogs. That would require a doubling of the shutter speed twice (1/60th to 1/125th to 1/250th) which would require two things to balance your exposure - either doubling the ISO twice (to ISO 20,000 +) or opening the lens up by 2 stops (f5.6 to f4 to f2.8 )
Doubling the ISO would result in a nasty noisy image.
Moving to f2.8 requires a big financial investment!
Now if you're shooting with an f1.4 lens you would then be able to shoot at f1.4 (4 stops faster) at 1/250th second and ISO1250 - this would give you a fantastic result on your D7000. That is IF your autofocus, in continual tracking mode, could keep up.
There's the advantage of "Fast" lenses. I'm not advocating going out to buy one straight away, but you've just set yourself a very challenging subject to try and shoot. Stick at it though!
Settings I'd select for this situation?
- Auto ISO ON (minimum shutter speed 1/250th second, max ISO 12800), then keep the camera in A mode and select wide open (f5 - f5.6) and hope!
Also worth considering - centre weighted or spot metering. Consider your autofocus tracking modes - I'm not sure on the D7000 but you'lll want some form of tracking on AF-C.
I'm not meaning this in a patronising way at all, but it may be a bit too much to ask straight off, shooting in this kind of environment. You're shooting in extreme conditions and there are lots of possibilities to work around your problem but it's all a balancing act with exposure. I'd getting a thorough understanding of exposure first before worrying about panning at 1/60th at night, getting motion blur from a panned background, with sharp subjects and blurry legs.
Wheel out that 15 year old arthritic cat in good light and start from there!