If the video messes up in "Speed" mode but works fine for long periods in "Stamina" mode, the video adapter is most likely overheating, especially given the appearance of the screen that you describe (which is typical of video adapter failures). If there are one or more fans in the laptop, verify that they are all turning. Verify that ventilation paths are not blocked by dust.
Laptops always have trouble staying cool, so intermittent video failures could be par for the course with the particular model you have, if it was poorly designed. One or more failures due to overheating may also produce permanent damage, causing intermittent failures even after the cooling issue is fixed.
Changing drivers will generally have no effect on cooling issues. The only exception would be if there's a variable-speed fan on the video adapter and the driver is not setting the speed appropriately, but that's highly unlikely.
You can find various helpful gadgets on the nVidia Web site, which may help you to measure the GPU temperature and (in some cases) set the fan speed, if a fan is present. They also have some demos on the site that you can run to show off the capabilities of nVidia cards, and some of these will pin the card against the wall in terms of video load, so they might be useful as a stress test. Watching the temperature may be the best thing to try, though. Video cards have a high tolerance for heating (more so than CPUs, in many cases), but there are still limits.