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 that deals with Gorenstein domains of codimension three. Here is a paper 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.