Tangents from a Line

The hyperbola $E \colon \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ ($a > 0$, $b > 0$) has a vertex on the line $l \colon y = x + 1$ and eccentricity $\sqrt{5}$.

1. Find the standard equation of the hyperbola $E$. 2. Call a line a tangent of $E$ if it meets $E$ in exactly one point and is not parallel to an asymptote; the common point is the point of tangency. Let $T$ be a point on $l$ from which exactly two tangents to $E$ can be drawn, touching $E$ at $P$ and $M$ respectively.

1. If $t$ is the $x$-coordinate of $T$, find the range of possible values of $t$.
2. Let lines $TP$ and $TM$ meet the line $x = -1$ at $Q$ and $N$ respectively. Prove that the intersection of lines $PN$ and $MQ$ lies on a fixed line.