Also, is there somewhere I can find a list of problems involving power series composition?

There are several ways to calculate the inverse of $$$x \exp(-x)$$$. Lagrange inversion theorem is the easiest one, but here's more tricky approach.

Let's consider the case of $$$A=(0)$$$ in this problem, then $$$\exp(-x) g(x \exp(-x)) = 1$$$. Let $$$h(x)=x g(x)$$$ and we have $$$h(x \exp(-x))=x$$$. This means that $$$h$$$ is the compositional inverse of $$$x \exp(-x)$$$. $$$[x^n] h(x) = [x^{n-1}] g(x)$$$, and considering the original problem with $$$A=(0)$$$, this equals the volume of the $$$(y_1,\cdots,y_{n-1})$$$ where $$$0 \leq y_1 \leq \cdots \leq y_{n-1} \leq n-1$$$ and $$$y_i \geq i-1$$$. Now we can calculate this value by hand and get the result of $$$n^{n-1}/n!$$$.