How many moments are needed for a discrete distributions
You can do $E\{ x^n \}$ as much as you want. How much is necessary is up to you but the unique characterization of a distribution is not the same thing as a cutoff point of a moment expansion.  Some distributions can be very compact with respect to their moments but others are not.  My experience is that moment (taylor like) expansions can converge very slowly. 

For something like a binomial distribution, all I need to know to represent the pdf is $p=$ success and know that the trials are independent.   The probability density for successes given a number of trials flollows from the mathematical formula.