The Largest Slope That Misses

Let $f(x) = x - 1 + \dfrac{a}{\mathrm{e}^x}$, where $a \in \mathbb{R}$ and $\mathrm{e}$ is the base of the natural logarithm.

1. If the tangent line to $y = f(x)$ at the point $(1, f(1))$ is parallel to the $x$-axis, find $a$. 2. Find the extreme values of $f(x)$, discussing all cases. 3. When $a = 1$, if the line $l \colon y = kx - 1$ has no point in common with the curve $y = f(x)$, find the maximum value of $k$.