That suggestion for the exponent is certainly in strong agreement with a linear regression on the log-log scale.

Am on the road, so don't have references handy, but just from memory there is no well-defined multiplication for games in general (the multiplication in On Numbers and Games works for an extended field of numbers, but not for all games).

I'd say more (and do so when I teach the material). There's really no point in learning anything but Myhill-Nerode which is both more general, and natural (though I hate the name for the usual reason that it says nothing about the result).

Sorry rather than "SW corner" that should be "W column" but the question still applies (if we put an entry in the NW corner, then the SW corner is still uncrossed, and if we put it in the SW corner then the NW corner is only crossed vertically).

I'm missing something. Consider the $2 \times 2$ case with $\pi$ just having a single 1 in the NE corner. The $\rho$ entries can only be in completely uncrossed boxes, i.e. in the SW corner alone, but this doesn't give a double crossing in the NW corner.

I'm not sure if the $z_{2n}$ identity is correctly explained (it is correct) -- it seems rather to follow from the fact that in general the sum of consecutive even and odd terms in $z_{2n}$ is equal to $(1 + e^x)$ times a single term contributing to $z_n$ rather than a matching between the first half and the second half. But perhaps I'm missing something.