If a particle is rigidly attached to a ring and the ring is undergoing pure rolling, can I take the the particle to be in pure rotational motion from the view point of the instantaneous axis of rotation? Why or why not?
Well, from the axis of rotation the particle is indeed only rotating. For instance if you are on a car and look at the tyres of another car driving at the same speed, if the the tyres are marked by a white spot, the white spot will look in pure rotation. This is why you can consider it in pure rotational motion : you have changed of referential.
But you must not forget that you are moving, therefore the axis is moving and therefore as you pointed it out it needs to be the instantaneous axis. The movement of the particle will be an cycloid.
This page : page and especially the gif which is on it might help you to see it