A Tangent-Bisected Focal Chord

The ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a>b>0$) has foci $F_1(-c,0)$ and $F_2(c,0)$ ($c>0$). A line $l$ through the right focus $F_2$ is tangent to the circle $x^2+y^2=b^2$ at a point $P$ and meets the ellipse at two points $A$ and $B$, with $A$ above the $x$-axis. Suppose the point of tangency $P$ is the midpoint of the segment $AF_2$.

1. Find the eccentricity of the ellipse. 2. Find the slope of the line $l$.