The stone faces and surrounding area is off limits to climbing. However, there is fair amount of quality rock climbing in the area, including bouldering, bolted sport routes, and trad routes up to 400 feet in length. The most current guidebook for the area is "Mt. Rushmore National Memorial Climber's Guide" by Vernon R. Phinney.
Essem's Answer is quality 1. We did a road trip there many years ago. It is a wonderful place to climb along with many other areas around there Sulvan Rocks and Devils tower come to mind. there were so many one pitch two routes! I bet you could have a life time of just climbing, just around there. As for Rushmore, even trying to climb the back of there heads is a federal climb.
No. It's a national monument, and therefore against the law.