An Exponential Dominates a Log

Let $f(x) = \mathrm{e}^x - 2ax - 1 + 2a$.

1. For $a \in \mathbb{R}$, discuss the monotonicity of $f(x)$. 2. Let $g(x) = (x - 1)\ln(x - 1)$. If $f(x) \geqslant g(x)$ holds for all $x$ in the common domain, find the range of possible values of $a$.