Solutions of $x^d=1$ in the symmetric group

Q2. Of course, this is a general thing for exponential generating functions. Assume that $a(n)$ is the number of ways to make lunch from $n$ distinct ingredients, $f(z)=\sum \frac{a(n)}{n!} z^n$ is an exponential generating function. Then, say, $f^2$ is an exponential generating function for making two enumerated lunchs from $n$ ingredients, $f^2/2$ for two not enumerated lunchs, $f^3/3!$ for three non-enumerated lunchs and so on. Totally $\exp(f)$ for arbitrary number of lunchs. In our situation lunch is a cyclic ordering of ingredients when the number of ingredients divides $d$.

For Q1, there is a quite nicely written 2014 preprint by Cheda and Gupta.