Do NASCAR race cars use automatic transmissions?
No, they do not. Manual transmissions are used in Nascar race cars.
No, automatic transmissions may shift faster than a manual transmission because it takes a lot of skill to race and change gears using a manual with a clutch, however, the manual transmission allows you to decide exactly when to shift for more speed at the times you need it and for better engine performance when done by a skilled, practiced, professional racer.
Yes, nascar race cars have a reverse. They are standard 4-speed Manual transmissions, in a not-so-standard car. They are anything but standard four-speeds. Everyone uses Jerico transmissions, which are made for ease of changing the gear cluster. If you want a different gear ratio in the car, you open the little door in the floorplan, open the top of the transmission, and do your work. Very simple and direct. They also definitely have reverse.