A nice summary of the status of these problems may be found here:

*Otal, Jean-Pierre*, **William P. Thurston: ``Three-dimensional manifolds, Kleinian groups and hyperbolic geometry"**, Jahresber. Dtsch. Math.-Ver. 116, No. 1, 3-20 (2014). ZBL1301.00035.

I would charaterize it this way: Problems 1-18 have been completely answered, although some of the answers are still unpublished (but exist as preprints and are still under submission). However, some of these questions (maybe 4., 7, and 8.) are less precisely formulated and somewhat open-ended, so one could argue whether or not they are answered completely. For problem 4., Hodgson's thesis addresses one part of the question: "Describe the limiting geometry
which occurs when hyperbolic Dehn surgery breaks down." I could list several other papers on this topic.

Problems 19-24 are more open-ended, and thus will likely never be completely satisfactorily addressed (I discussed some of these problems here). No progress has been made on problem 23 as far as I know (which was due to Milnor originally, not Thurston). One can imagine,
however, a complete and satisfactory answer eventually to question 23.

Otal does not comment much on Problem 24, which again is somewhat imprecise (what does "most" mean?). As Igor mentions, Joseph Maher has given a satisfactory answer. One could also argue that this was satisfactorily answered in Hempel's paper together with geometrization (Heegaard distance $>2$ implies hyperbolic). But there is other work on this question, such as giving a model for a hyperbolic manifold of bounded genus and bounded geometry (a lower bound on the injectivity radius) by Brock, Minsky, Namazi, and Souto. Moreover, these authors have a program to understand the unbounded geometry case as well (so it would eventually give a description of the geometry of all hyperbolic manifolds of bounded Heegaard genus in some sense). Thus, one could consider this problem to still be open for a variety of reasons.