Two Incircles of Focal Triangles

Let $F_1, F_2$ be the left and right foci of the hyperbola $x^2 - \dfrac{y^2}{3} = 1$. A line through $F_2$ with inclination angle $\theta$ meets the right branch at two points $A$ and $B$. Let $r_1, r_2$ be the inradii of triangles $AF_1F_2$ and $BF_1F_2$, and $S_1, S_2$ the areas of the corresponding incircles. Determine, with proof, which of the following are true:

1. $\theta$ ranges over $\left(\dfrac{\pi}{6}, \dfrac{5\pi}{6}\right)$; 2. the line joining the two incircle centers is perpendicular to the $x$-axis; 3. if $r_1 + r_2 = 2$, then $|AB| = 6$; 4. $S_1 + S_2$ ranges over $\left[2\pi, \dfrac{10\pi}{3}\right)$.