Here is another recent thread discussing the combinatorics. Some numbers and references are provided within the first few posts.
What exactly do you mean by this? Do you mean “number of legal moves” at each turn? Or something more narrow, like “number of reasonable moves” at each turn?
The first is rather easy to compute given a game, and one could average those curves over several games to get a general idea. The second is a bit trickier to define, but perhaps one could use a bot to judge how many moves are reasonable at each turn (like by only couting those that do not lose too many points).