Given an infinitely differentiable function $ g: \mathbb{R} \rightarrow \mathbb{R}$, can we always find a distribution function $f_X$ of some random variable $X$ so that $g(t) = \int_{-\infty}^\infty e^{tx}f_X(x) dx$?
If my question is too vague or ill-posed, can anyone recommend any literature on the characterization of moment-generating functions?
No, there are a few necessary conditions, I don't know if there's a sufficient one.
If $g(t) = \mathbb E(e^{tX})$ we must have $g(0) = 1$ and $\lim_{t\to -\infty}g(t) = \mathbb P(X=0) \in [0,1]$.
we also have $\mathbb E(X) = f'(0)$ and Var$(X) = f''(0) - f'(0)^2 >0$. We also have by Jensen's inequality
$$\begin{array}{rl}g(t) &=\mathbb E(e^{tX}) \\&\geq e^{t\mathbb E(X)} \\ &= e^{tf'(0)}\end{array}.$$
So there are lots of probabilistic statements you can make that must be satisfied by anything that's a probability generating function. It's quite easy to find counterexamples.