Two Functions, One Parameter

Let $f(x) = \ln x - ax$ and $g(x) = \mathrm{e}^x - ax$, where $a$ is a real number.

1. If $f(x)$ is decreasing on $(1, +\infty)$ and $g(x)$ has a minimum value on $(1, +\infty)$, find the range of possible values of $a$. 2. If $g(x)$ is increasing on $(-1, +\infty)$, determine the number of zeros of $f(x)$, and prove your conclusion.