Probability of all alleles represented in a sample
I'm going to strictly answer the questions, rather than step through the proof, because it involves a lot of formatting that I'm not familiar with.  Other folks are welcome to edit this!
Equation 13
This equation assumes a diploid genotype, given by the $2n$ power with $n$ individuals.  For anything with greater ploidy than mono-, it's mathematically simpler to determine the probability that an allele is not present.  As an example, see this calculation of a triploid Hardy-Weinburg equilibrium equation.  Using this simplification,
$P(single$ $allele$ $not$ $present)$ $= (1$ $- P(allele$ $present))$ ^ $(ploidy)$ ^ $(n)$
$= (1$ $- P(allele$ $present))$ ^ $(ploidy$ * $n)$
With $k$ segregating alleles, each allele has its own non-presence probability.  The probability of total non-presence is $1 - (sum$ $of$ $P(each$ $non$-$presence))$
Equation 16
In this equation, the author describes the probability that all alleles are present at a given frequency.  These allele presences are independent of each other and therefore multiplicative.  Since $P(allele$ $present)$ is vectorized, this product can be simplified to $^r$