Do you have VSync enabled in the catalyst driver settings? If not, do so. For that black pixel thing, could be that your card gets overheated sometimes. Download
GPU-Z and tell me the idle temperature(when you don't do anything) and then play a graphic intensive game for like 5 minutes and tell me then the temps(you need to check the box where it says "Continue refreshing this screen while GPU-Z is in the background").
If you're temperatures get up to 80°C, check the air flow in your computer. Do you got a fan in the front and one on the back(or side)? If not, make sure to get them. 120mm if possible, since they are more effective while being not that loud as an 80mm fan. Also try to keep any cables out of the way. I don't know how your case is build up, a picture of the inside would be nice, so I can help you get a decent air flow.
An alternative would be to download the omega drivers. That helped me with my old X1800GTO, atleast for 2 months, then it finally fucked up and I had to buy a new one, due to overheating.
