I don't think these are problems with the actual controller.
As far as your second problem - In black ops 2 there are a few bugs. One of them I've noticed is if you move your analog stick while the game map is loading you'll turn in circles and won't be able to stop. if you are on a menu you'll scroll through the menu uncontrollably. You'll have to switch to a different controller then switch back to fix it. I've experienced this numerous times and it seems to be tied to moving the analog stick while the game is loading. This could be what's happening to you. Next time it happens try switching to a different controller then back to the gamepad and see if it fixes it.
Also, your first problem sounds like game lag that's related to your TV. Play windwaker for a minute on your gamepad screen only - off of your TV, see if it lags. If it doesn't, then it's for sure a problem with your TV. Have you turned your TV to "game mode"? My TV will lag as you described if I'm not in game mode. Check your TV's settings. Also, what TV are you using? if you are using an older TV or a cheap TV it might not have the specs to keep up with the console.
If these don't work you could try resyncing your gamepad with the console.


