Recall that I admitted I was wrong in my first posts on 4 and 13. In 7 there is an impulsive (= abrupt) change in angular momentum. Here we have to choose the reference point carefully even for what happens immediately afterwards.
In 13, there is no abrupt change, and we are only concerned with what happens immediately on release from rest. That means that everywhere, even parts of the bodies, are effectively stationary points and can be used as reference axes. Your original method didn't work because you forgot the torque from friction. (Note that when there is an unknown force that you don't need to determine to answer the question, you can often sidestep it by taking moments about a point on its line of action. You could instead take moments somewhere else and combine that with a linear equation to eliminate the unknown force. The two methods produce the same answer, but the first can be quicker.)
In 4, there is no abrupt change, but we are interested in what happens some time after the 'event', so again we need to be careful. [When we talk of a collision we usually mean that there are large forces that act for a short time. We don't know (and perhaps don't care) how large is the force or how short the time - we only only know the momentum imparted, ∫F.dt.
In 4 there is no collision in this sense. It's all pretty smooth.]
I think I mentioned on some thread that there are essentially two safe choices for taking moments: any fixed point or the mass centre of the system. In 4, (after rereading the problem statement), the midpoint of the rod is the mass centre of the system. It is OK because the linear momentum of the system has no angular moment about the mass centre.