A Parabola and a Min Triangle Area

The parabola $C:y^2=2px$ ($p>0$) has focus $F$. Let $A$ be any point on $C$ other than the origin; a line $l$ through $A$ meets $C$ again at $B$ and meets the positive $x$-axis at $D$, with $|FA|=|FD|$. When the $x$-coordinate of $A$ is $3$, $\triangle ADF$ is equilateral.

1. Find the equation of $C$. 2. Let $l_1\parallel l$ have exactly one common point $E$ with $C$. (a) Prove that the line $AE$ passes through a fixed point, and find it. (b) Does $\triangle ABE$ have a minimum area? If so, find it; if not, explain why.