A Tangent, an Extremum, and a Bound

Let $a > 0$ and $f(x) = ax - x\mathrm{e}^x$.

1. Find the equation of the tangent line to the curve $y = f(x)$ at the point $(0, f(0))$. 2. Prove that $f(x)$ has a unique extreme point. 3. If there exists $a$ such that $f(x) \leqslant a + b$ for all $x \in \mathbb{R}$, find the range of possible values of the real number $b$.