An Isosceles Configuration

Consider the hyperbola $\Gamma \colon x^2 - \dfrac{y^2}{b^2} = 1$ (with $b > 0$) with left and right vertices $A_1, A_2$. A line $l$ through the point $M(-2, 0)$ meets $\Gamma$ at two points $P$ and $Q$, with $P$ in the first quadrant.

1. If the eccentricity is $e = 2$, find $b$. 2. If $b = \dfrac{2\sqrt{6}}{3}$ and triangle $MA_2P$ is isosceles, find the coordinates of $P$. 3. The extension of $OQ$ meets $\Gamma$ at a point $R$. If $\overrightarrow{A_1R} \cdot \overrightarrow{A_2P} = 1$, find the range of possible values of $b$.