Understanding gyroscopes Considering the typical situation of a rotating bicycle wheel held by one end of its axle by a rope tied to the ceiling: gravity torque is the time derivative of the angular momentum, and in this case is perpendicular to the angular momentum vector which makes it revolve without changing its magnitude (it precesses). 
My question is the following: the wheel stops turning eventually because of the friction in the string (it also slows down its own rotation because of friction in the bearing but let's ignore it), but could you clarify how, step by step? Is it because the gyroscope creates a torque on the string axis, but it's reduced by the friction torque, and as a result the friction torque translates back into a residual "gravity torque" (in other terms, the gyroscope "gives up" on a certain amount of gravity torque it can't counteract because of the string)?
To progress toward my actual problem in several dimensions, what if the friction torque is in fact torque given by another gyroscope on the same solid? 
 A: I'm guessing that you understand clearly the effect of precession here. The reason why the wheel starts to fall down is that when we explain the change in angular momentum of the wheel, we say that the angular momentum vector only changes in direction- right? But the angular momentum vector of the wheel doesn't only point outwards; it also points upwards (it has an upwards and an outwards component). The outwards component is the one that most people keep in mind- it's due to the rotation of the wheel about its center. But the upwards component of angular momentum is due to the rotation of the wheel about the rope. 
When considering the torque on the wheel, we can assume that the only thing the torque will do is change the direction of the outwards component of angular momentum (when there's no friction). So, we 'blame' this change of angular momentum on the torque exerted by the wheel's weight, and everything's fine. However, due to friction, the wheel's rotation about the rope slows down, resulting in a change in magnitude of the upwards-pointing component of angular momentum. This equates to a downwards pointing torque on the wheel, which will make it begin to fall. 
I had a lot of trouble understanding the phrasing in your question- I hope this is what you were doubtful about. As for your second questions, I have no idea what you meant ("What if the friction torque is in fact torque given by another gyroscope on the same solid?").
Hopefully I was of some help, though. 
