Yes, the grid fins can control the rotation of the stage -- and they're the primary mechanism for that during descent, since the cold-gas thrusters have little force and the engines are shut down for most of the time.
To keep the stage stable, the fins are left horizontal, and adjusted by small amounts to correct for other sources of rotation.
To spin the stage, all the fins are tilted in the same direction relative to their mounting point, so that opposing pairs are producing a horizontal force in opposite directions.
SpaceX accidentally tested your last question on the CRS-16 mission -- all the fins got stuck near the limit stop due to a hydraulic problem. The answer is: quite fast!