Perhaps this is too late for your seminar, but there has been a huge literature on these problems and related ones. The key words to search are: "h-vector of...". For example, here is a [paper](https://projecteuclid.org/euclid.nmj/1118775397) that deals with Gorenstein domains of codimension three. Here is a [paper](http://www.combinatorics.org/ojs/index.php/eljc/article/view/v11i2n4) on h-vector of Gorenstein toric ring, which Section 4 of Stanley's paper addressed.  

In general, even for Artinian Gorenstein algebras, a complete chracterization of h-vector seems to be out of reach, but there are many surveys and works are being posted at [this very moment](https://arxiv.org/abs/1804.08745).