A Hyperbola from a Geometric-Mean Condition

Let $A$ be the right vertex of the hyperbola $C:x^2-\dfrac{y^2}{b^2}=1$ ($b>0$). A line $l$ through the point $T(0,t)$ meets the left and right branches of $C$ at points $M$ and $N$ respectively.

1. If the slope of $l$ is $2$, find the range of $b$. 2. The lines $AM$ and $AN$ meet the $y$-axis at points $P$ and $Q$ respectively. If $|AT|^2=|PT|\cdot|QT|$, find the equation of the hyperbola $C$.