The product of any k characters is again a character. How do you get anything from joint moments beyond what happens for single characters?

I don't have a reference to hand for Singer's result - it is simply about the existence of an element of order $q^{n}-1$ in $\GL(n,q)$. This seems to be what you are referring to - it applies only when $q$ is prime power. If you know of an example of a projective plane where $q$ is not a power of a prime, you should publish it. You'll be quite famous!

Note that taking any subset of {1..n} in your first column and any fixed point free permutation, you can generate a latin rectangle. The block design doesn't really play any part... I guess you could characterise the rectangles arising from designs in terms of how often pairs of elements appear in each column, but that's essentially just a translation of the definition of the design. I can't see that they have any other special properties.

Among symmetric designs, these are exactly the ones which correspond to difference sets. You should look at Beth, Jungnickel and Lenz for a good introduction to this topic. For non-symmetric designs, I don't know of a good single reference. But google turns up many results for 'block transitive designs'.

You could look at "On a Theorem of Frobenius: solutions of $x^n = 1$ in finite groups" by Isaacs and Robinson. On the way to proving Frobenius Theorem, they show that $\phi(n) \mid s_{n}$ for all $n$, but some of the methods and references might be of use. I assume you want to exclude the case that there are no elements of order $2p$ in $G$, in which case $2p \mid 0$? (E.g. elements of order 6 in $A_{4}$)

Zsigmondy's theorem shows that, for fixed p and any value of n, there exists a prime dividing $p^n-1$ which does not divide $p^m-1$ for any $m < n$. en.wikipedia.org/wiki/Zsigmondy's_theorem