A Tangent and an Intersection

Let $f(x) = (x + t)\ln x$, and suppose the tangent line to $f(x)$ at $x = 1$ is parallel to the line $x - y = 0$.

1. Find $t$, and find the intervals on which $f(x)$ is monotonic. 2. Let $a > 0$. If the graphs of $y = \mathrm{e}^{ax}$ and $y = \dfrac{f\left(\frac{1}{x}\right)}{ax}$ intersect for some $x \in \left(0, \dfrac{1}{\mathrm{e}}\right)$, find the range of possible values of $a$.