I think it is because its area of contact with the oxidizing agent increases with the size reduction (when you compare samples with the same mass and different particle sizes).
I think so. and a few methods make the expansion of the graphite with acids to increase the distance between the graphite sheets and improve oxidation step.
Graphite, or graphene, is reactive by the edges. Without edsges, the reactivity would be extremely low, only controlled by defects in the plane (such as terrace borders). But edges behave as defects, bearing dangling bonds. So these parts are very easy to oxidise, as carbon aims at satisfying a 4th bond. Therefore, the lower the particle size, the higher the contribution of edges. This explains the higher reactivity.