My question is: is it known whether estimating the size of the convex hull is also lower bounded by n log n (or n log h) in this model ? It seems like this should be true: I don't see an obvious way of using a size-estimation oracle to actually compute the hull (or its vertices) in linear time, but I don't know if there's a proof of this. Yao's proof doesn't appear to yield any direct insight into this question.
It's worth pointing out in the light of recent results by Chan, (and by Pătraşcu on point location), that we can solve the 2D convex hull problem in sub- n log n time; we pick our favorite machine model and sort the x-coordinates in sub-n log n time, and then run the Graham scan in linear time. Chan's paper achieves sub n log n bounds for the 3D convex hull and related problems.