When are growth series rational?
An answer to your second question: Stoll constructed many 2-step nilpotent groups such that there exist generating sets $S$ and $S'$ such that the growth series with respect to $S$ is rational and with respect to $S'$ is transcendental. See
M. Stoll, Rational and transcendental growth series for the higher Heisenberg groups, Invent. Math. 126 (1) (1996), 85–109.
As for your first question, I don't know if it is intrinsically interesting, and I am not aware of any applications of these results. However, what is interesting are the vast array of tools and techniques that are used to study it. It is a good proving ground for many ideas in geometric group theory (especially those related to things like regular languages). There is an enormous literature on this topic. The introduction to my paper
A. Putman, The rationality of sol manifolds J. Algebra 304 (1) (2006) 190-215.
summarizes most of the papers concerning it that I am aware of. It can be downloaded from my webpage here. The only ones I know about that came out after it are
My student Corey Bregman's paper "Rational Growth and Almost Convexity of Higher-Dimensional Torus Bundles", available here.
Duchin-Shapiro's paper "Rational growth in the Heisenberg group", available here.
I don't know how interesting this is to you, but knowing that $S(z)$ is a rational function is a quick way to show that the exponential growth rate $\lim_{n \rightarrow \infty} \sigma(n)^{1/n}$ of the group with respect to $S$ is an integer algebraic number. So, for example, the only exponential growth rate you can get out of a hyperbolic group is an algebraic number.
I'll note that there do exist exponential growth rates that are not algebraic numbers--in fact, there are uncountably many possible exponential growth rates, as shown by Anna Erschler in
A. Erschler. Growth rates of small cancellation groups. In Random walks and geometry, pages 421–430. Walter de Gruyter GmbH & Co. KG, Berlin, 2004.