Minor correction: the lower Hamming bound for $g=5$ if exact not only if one of the $a_i$ is 1, but also in the particular case where all the $a_i$ are equal to 2.
Good synthesis of various results, by the way, thanx!

Also, on the problem of counting rhombus tilings for $d>3$ (for $d=3$ there are some beautiful results by Kenyon, Propp, Larsen, Cohn & Others), some partial results have been obtained by Nicolas Destainville (in particular in the $d=4$ case), see e.g.:
http://www.springerlink.com/content/l0l582625616131n/