The point is that conservation principles are not generally intuitive. For example, why should energy be conserved? One must have a grip of the dynamics involved in order to understand them. 

Anyway, the precession of the spinning top doesn't have to do with the conservation of angular momentum. It has to do with the strange nature of torque and its interaction with angular momentum. When a force acts on a spinning top, it excerpts a torque perpendicular to the plane defined by the axis of the top and the direction of gravity, which is a vertical plane. That direction is horizontal. 
On the other hand, the torque is the rate of change of angular momentum. That means that the direction of the torque is the direction towards which the vector of angular momentum changes. Thus, since the torque is horizontal and perpendicular to the angular momentum, it can only change the direction of angular momentum along the horizontal direction and not towards the ground. That means that the vector of angular momentum has its back on the ground, at the point that the tip of the top touches the ground, and its head is performing a circle on a plane that is parallel to the ground. That motion is the precession of the spinning top.   

Finally, I think that the reason for assuming a much faster rotation than precession for the top, is to simplify the calculations and consider the top as a gyroscope. 

Why don't spinning tops fall over?
I can't understand the relation between Schrödinger's Cat and this amazing experiment of the double slit. 

It seems like in the double slit we know if we observe the particle the wave function collapse and the electron acts like a normal particle, so doesn't that solve the problem in Schrödinger's Cat?

When we open the box the atom that should help release the poison acts like a particle,the poison is in the bottle so the cat stays alive, don't know really?!
