At least partial answers to both your questions may be found in the following [paper][1]:

> Patricia Hersh and Isabella Novik, A short simplicial $h$-vector and the upper bound theorem, *Discrete & Computational Geometry*, 28 (3): 283-289 (2002).

In particular, the main theorem implies (via Corollary 1) that if the dimension $d$ is odd, then the first $d-1$ entries of your $f$-vector will be bounded above by those of the [cyclic polytope][2] $C(n,d)$. For additional details, look for the "upper bound conjecture" as well as the references in this paper.

I don't think much is known when $d$ is even.


  [1]: http://arxiv.org/pdf/math/0111302
  [2]: http://en.wikipedia.org/wiki/Cyclic_polytope