An Integer Slope Ceiling

Let $f(x) = \mathrm{e}^x\left(x\ln x + \dfrac{2}{\mathrm{e}}\right)$ and $g(x) = ax$ with $a \in \mathbb{Z}$, where $\mathrm{e}$ is the base of the natural logarithm.

1. Find the equation of the tangent line to $f(x)$ at $x = 1$. 2. If $f(x) > g(x)$ holds for all $x > 0$, find the maximum value of $a$.