Is there an example of an elementary function (different from Dirichlet polynomials, i.e. cutoff Dirichlet series) which has a know Dirichlet expansion (known coefficients)?
I am aware of the coefficient forumla but it does not seem to be practical for elementary functions.
It seems that the Dirichlet series are specific to the number theory and completely avoid the domain of mathematical analysis, with the exception of the Zeta function (which is however not elementary).
Dirichlet series do pop up in analysis, specifically in basis theory: Hedenmalm, Lindqvist and Seip used them to solve the "dilation basis problem" in $L^2[0,1]$ in this paper: https://arxiv.org/abs/math/9512211.
To answer your first question: you can rewrite any function f(z) that is analytic on B(0,1), i.e. the open disk with radius 1, as a Dirichlet series g(s) defined on the open half-plane $Rs > 0$ through the change of variable $z = p^{-s}$, for some natural number $p$.