I'm pretty hopeless with any non-decimal base, but bases above ten are the stuff of nightmares for me. Still, once I got the hang of things using an internet hex/decimal converter, getting most of the entries was more a question of logic than maths. I think I have the right solution, subject to checking, but the 3- and 4-letter words didn't work out as I think they should have, so I need to play the game again. As I have a 9-letter word that I can see is thematic in a surprising way, I think my grid is correct.
I think the preamble is very misleading, if not inaccurate, in saying, "Having deduced all entries, solvers must play out a game of Solitaire." It is not possible to deduce all entries from the clues alone, and the ambiguities can only be resolved by playing the game. In anticipation of the penultimate sentence of the preamble, the fourth sentence should have said something like, "Having deduced all entries as far as possible, solvers etc."
I know we're supposed to read all the preamble first, but in a puzzle such as this, with a very complex set of instructions for playing the game, I'm sure I wasn't the only one to defer reading past the fourth sentence in the interest of getting the grid filled first, before getting to grips with the game's procedures. An early warning of ambiguities was necessary in my view.
If anyone is still working on the puzzle, they might these two links useful:
http://www.rapidtables.com/convert/number/hex-to-decimal.htm
http://www.csgnetwork.com/hexmultdivcalc.html
The first is a conversion program (there's also a parallel decimal to hex page); the second can save some time working through the choices that are divisible by 6 or 7.