Well, for this door situation, it wasn't necessarily stated that it is a rotating door (it could be a sliding door instead). But I will assume it is a rotating door.
A) Yes, the doorknob does experience centripetal acceleration. It is traveling at a constant speed, but it is changing direction. Changing direction while still moving, requires a centripetal acceleration, and is directed toward the door hinge (hence the name).
B) No, The doorknob is a fixed distance from the stationary hinge. The doorknob moves at a constant speed. No change in speed means no acceleration tangent to the path of motion.
C) No. There is no centripetal acceleration, because the doorknob is instantaneously stationary when open at its maximum. Zero velocity = zero centripetal acceleration (formula is a=v^2/r, and v=0) required to change the direction. All acceleration at this point is in the direction of the doorknob's travel direction, and not perpendicular toward the hinge.
D) Yes. The doorknob is reversing at this moment. It is slowing down, then speeding up. The doorknob is undergoing acceleration along its direction of motion, which is the meaning of tangential acceleration.