This thread seems to be unravelling somewhat, so I thought I'd clear up one part of it now: I have confirmed Bertram's calculation for the number of valid Sudoku grids.
I've actually done this in two different ways:
1. fix top-left 3x3 contents + fix position of 1s (as suggested above and ages ago by someone else). This leaves 2000+ "templates" for the positions of each of the digits 2 to 8, each of which can be represented as a 64-bit int. Grind through these recursively. Takes ages, but works, and simple enough that it's hard to argue with.
2. fix representative contents of top 3 rows (27 cells). This is essentially Frazer's method. I can get this down to 44 representatives (Frazer and Bertram had 71 but noticed there were only 44 different results coming out of those). For each representative, solve the rest of the grid:
(a) by just chucking down 64-bit templates
or, (b) by looping over representative contents of 3x3 boxes 4,7 + position of remaining 1s, then solving remainder with 32-bit templates [sorry, I know this isn't making sense]
Each representative takes a couple of minutes to count and the result is the same as Bertram's and method 1 above.
Bertram has added my code to his archive if anyone wants to check it.
Now we just have the minimal grid question remaining on this thread. I think that's going to be quite a lot harder.