An Area-Doubling Line

The ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a>b>0$) has left and right vertices $A_1$, $A_2$ and right focus $F$, with $|A_1F|=3$ and $|A_2F|=1$.

1. Find the equation of the ellipse and its eccentricity. 2. Let $P$ be a moving point on the ellipse (not a vertex), and let the line $A_2P$ meet the $y$-axis at a point $Q$. If the area of $\triangle A_1PQ$ is twice the area of $\triangle A_2FP$, find the equation of the line $A_2P$.