Ok... I played with GAP today. Here is what I produced:
If needed here are some definitions:
Normal = All type 1 Faces
Circle = All type 0 Faces
Crazy Plus 3x3x3 = Normal Puzzles as defined above.
Super Crazy Plus 3x3x3 = A Crazy Plus 3x3x3 where each pieces has a unique position AND orientation in the solved state. Face center orientation is considered.
DoDep 3x3x3 = My DoDep series stickered with normal stickers.
Super DoDep 3x3x3 = My DoDep series stickered with super stickers
Multi DoDep 3x3x3 = A DoDep puzzle viewed as a Multi-5x5x5 (with a 3x3x3 inside). Normal stickers applied to the 5x5x5 and the 3x3x3 inside.
A Super-Multi DoDep 3x3x3 = A DoDep puzzle viewed as a Multi-5x5x5 (with a 3x3x3 inside). Super stickers applied to the 5x5x5 and the 3x3x3 inside.
For the DoDep calculations I also included a 5x5x5 for comparison. For example the Super-Multi 5x5x5 would be my Real5x5x5 puzzle
Some interesting observations.
(1) Except for the Venus and Mercury versions the Normal DoDep puzzles have the same number of permutations as a Normal 5x5x5.
(2) Except for the Venus and Mercury versions the Super DoDep puzzles have HALF the number of permutations of a Super 5x5x5. So there is an interesting added parity to the face centers as I suspected.
(3) Only the Jupiter Multi-DoDep puzzle has the same number of permutations as a Multi-5x5x5. This is due to it being the only version that allows you to fully scramble the inner 3x3x3.
(4) The Jupiter Super-Multi DoDep puzzle comes closest to the number of permutations of the Super-Multi 5x5x5 but it is still a factor of 4096 short.
I've attached the GAP files and a text file containing the above numbers. If anyone wants to double check the calculations please let me know if you get the same numbers.