Ma'am I didnt mean to post as a correction. I believe the question is poorly worded. Here is the issue, the monkey does not change the tension of the rope if he is moving at a constant speed. Therefore if he is smooth, constant and uses all 4 hands, he could theoretically get to the top without changing the tension on the rope for more then an instant (initial acceleration). The friction of the pully and the air resistance of the system throws a larger kink into it, however, tension as a force cannot be effected by a constant velocity movement.
These are the facts, everything else is a matter of interpretation:
The monkey's position on the rope is completely irrelivant to the equilibrium to the system, only that the moment arm on the pulley is the same for the weight and the monkey.
As he accelerates, he tensions the rope. Though the mere fact of him moving (if at a constant velocity) does not effect the tension of the rope. Gravity pulls on him the same everywhere on the rope, so the force felt by the rope does not change.
I dont know what the Navy says the answer is, and that is the only thing that can truely settle this. I see your reasoning, and find it sound and understandable, however I disagree with the specifics. The assumptions required to analyze this to the degree that we are, are more numerous then the problem warrents. The monkey position is irrelivant, I think that is the point of the question, not that he could tension the rope when he climbs.
I put c and didnt do too bad. If someone is going to see their recruiter soon, see if you can get it from him. The ROTC unit here doesnt have the answer key.
P.S. does anyone have a monkey so we can try this?