Consider the forces on the rider: there are three.
1. g, acting downward; acceleration g = 9.8 m/s²
2. centifugal force (call it C), acting outward; accleration C = v²/R
3. The force on the cable (call it T), which MUST close the triangle at the hypotenuse, hence T = sqrt (g² + (v²/R)²)
Then θ...
Your friend is right. The climber's weight is only one of the two forces acting on the rope. The second is the force caused by the upward acceleration of the climber. According to Newton's Second Law
f = m a
Force = Mass x Acceleration
Since the climber isn't static -- he's moving...
When r=0, you no longer have two masses, you have one. This condition actually does occur at a black hole, where all mass is (believed to be) contained in a singularity, i.e., a single point.