Progress on generalized star-height problem?
Regarding your second question, an explanation why the generalized star height problem is less accessible than the star height problem is the following: Already Eggan's seminal paper in 1963 contained languages of (ordinary) star height $k$, for each $k\ge 0$. Only a few years later, McNaughton, and, independently, Déjean and Schützenberger, found examples over binary alphabets. This made clear what the problem "is about". During the years that followed, there was a more or less steady flow of published results in the area of the ordinary star height problem. This gave an ever increasing body of published examples, counterexamples and phenomena surrounding this problem.

In contrast, after some fifty years now, we don't know whether there is any regular language of star height at least two. So we do not even know whether there is a need for a decision procedure after all. This "complete lack of examples" indicates that it is extremely difficult to get a grip on this problem.