What sort of friction does a rolling cylinder experience? When will static friction act on a cylinder executing pure rolling? Will it act  when there is some external force (or without it) and if yes wouldn't it rather be kinetic friction?
The cylinder is rigid, so no deformations and I'm neglecting air resistance.
 A: Yes, static friction acts on the rolling cylinder only when there is some external force. Then friction and the external force create a torque causing acceleration or deceleration. 
When the cylinder is rolling with constant velocity the static friction force is zero; it does no work on the cylinder. It would make no difference if the plane on which the cylinder is rolling were to become frictionless in this condition : the change would not alter the angular or translational velocities of the cylinder, because friction is not required to keep the cylinder rolling.   
Kinetic friction only acts when there is some sliding - ie when the no slip condition $v=r\omega$ is not met. Then there is relative motion between the cylinder and the plane at the point of contact.
A: The definition of pure rolling as based on wikipedia is "Rolling where there is no sliding is referred to as pure rolling" but the more technical definition for pure rolling would be "when there is no relative motion between the point at which the cylinder touches the surface and the surface in itself"; for kinetic friction to happen there must be some relative velocity between the two surfaces mentioned. Thus, pure rolling will experience static friction rather than kinetic friction. 
A: When a disk's motion is purely rolling, that means that the contact point of the disk and the ground are in relative rest.
This is the definition of "pure rolling",and put another way, it means that the part of the disk in contact with the ground is in rest with respect to the ground, and because of that, all friction that the disk experiments is always static friction, not kinetic friction. 
