Why pure rolling's energy is conserved? Although there exists non-conservative force during pure rolling (for this, static friction), why is energy conserved? Is it just special feature of pure rolling?
 A: I am not sure static friction is considered a non-conservative force at all. Nor conservative. The definition from Wikipedia:

A conservative force is a force with the property that the total work done in moving a particle between two points is independent of the taken path

Static friction is by its very nature unable to move anything anywhere. It is keeping the contact point stationary - nothing moves where the force acts. The rest of the ball moves, yes, but not the contact point.
So static friction does no work (work requires a distance moved) and is not involved in this definition of conservative or non-conservative forces. A ball rolling is influenced by static friction as the force that start rotation, but only other forces such as the weight (which is conservative) can do work.
A: Work is associated with force and displacement. When a body rolls on a horizontal plane at a constant velocity, there is no external force. Since the point of contact with the surface has zero velocity relative to the surface therefore the static friction does no work.Irrespective of the nature of surface, friction is zero. No work, therefore, is done on the body. There is no corresponding change in the kinetic energy of the rolling body.
For an accelerated rolling, an external force is applied on the body. Application of external force in rolling, however, is not very straight forward as in the case of translation. Static friction comes into picture – whenever there is sliding tendency. Depending on the situation, friction plays specific role to maintain rolling.
A: Energy is conserved simply because STATIC friction acts on the body.Static friction does no work.
Consider a thin disc rolling down an inclined plane.Friction acts on the bottom most point of contact which is instantaneously at REST.As no displacement is done while the force acts on the point of contact,work done by friction is zero.
Now by Work energy theorem we know that Change in Kinetic Energy=Work done by all the forces.As now we have established that only conservative force do work now,Change in kinetic energy=Work done by conservative forces which means mechanical energy in total remains conserved.
