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max(p_k) for sufficiently large k. Else, estimate assuming a binomial distribution with P(X=1) for max(p_k) i.e p=1/n. Mean would be k/n. If (k/max(p_k)) estimates the number of observations max(p_k) then n = ~max(p_k)
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If I understand your question correctly, yes. Use Beta updating to estimate the possible proportion of each side after each roll. I have a short tutorial on this.
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If not sequentially numbered, I wonder if you could look at it like the same birthday problem in reverse: if you have x pairs of people with the same birthday (pairs of rolls where p_A = p_B) in a room of k people, how many days are there in a year?
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