Re: normal distribution
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Using the naive bound on the 3rd moment, which we get using the fact that the amount you can win in a hand is bounded, Berry-Esseen says we're not even close to a normal until 100K hands.
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Is the problem the Berry-Esséen theorem, or the bad estimate you are using for the 3rd moment? I would not be surprised if the Berry-Esséen estimate could be improved in a more restricted context, but I think you may get a significant improvement by estimating the 3rd moment from actual data.
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The problem is definitely with Berry-Esseen, because the problem remains if you use the SD as the value of the 3rd moment. And by Jensen's inequality, the true 3rd moment (i.e. the cube root of rho, in the notation of the link you provided) is always greater than or equal to the SD.
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