The paper "Problems  on  billiards,  flat  surfaces  and  translation surfaces" is published in collection "Problems on Mapping Class Groups and Related Topics", edited by B.Farb,  Proc. Symp. Pure Math., Amer. Math. Soc., 233-243, 2006
(see also the web-page of Benson Farb, the Editor, for the online version of the entire collection).

Several related problems can be found in the last paragraph of the long survey A.Zorich, Flat surfaces, in collection "Frontiers in Number Theory, Physics and Geometry. Volume 1: On random matrices, zeta functions and dynamical systems,  P. Cartier; B. Julia; P. Moussa; P. Vanhove (Editors), Springer-Verlag, Berlin, 2006, 439-586.

Progress since the paper was published:

Problem 1: In particular case, when a flat surface is a tetrahedron  (a sphere with four conical points) closed geodesics are studied in  the paper:  V.Yu. Protassov, Closed geodesics on the surface of a symplex, Sb.Mathematics (Matematicheskii Sbornik), 198 (2007), No 2, 103-121

Problem 15. First results are obtained in the paper:  M.Kontsevich, Lyapunov   exponents   and   Hodge  theory, "The mathematical beauty of  physics" (Saclay, 1996), (in Honor of C. Itzykson)  318-332,  Adv.  Ser.  Math.  Phys.,  24.
World  Sci. Publishing, River Edge, NJ (1997) (see also arXiv).

Some results for Veech surfaces are obtained in the paper: math.AG/0511738 Irene I. Bouw, Martin Moeller,  Teichmueller curves, triangle groups, and Lyapunov exponents.

Genus 2 is studied in the paper: math.GT/0611409 Matt Bainbridge Euler characteristics of Teichmüller curves in genus two.178 pages.

Problem 17. Some interesting examples are constructed in the papers:  J. Smillie, B. Weiss, Veech dichotomy and the lattice property, preprint  2006;  and in the paper : math.DS/0607179 Yitwah Cheung, Pascal Hubert , Howard Masur, Topological dichotomy and strict ergodicity for translation surfaces.