A Fixed Point and an Area Chord

The ellipse $C:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a>b>0$) has right focus $F$. A line $l$ through $F$ meets $C$ at points $A$, $B$; the line through $F$ perpendicular to $l$ meets $C$ at points $D$, $E$, with $B$, $D$ above the $x$-axis. Let $M$, $N$ be the midpoints of $AB$, $DE$. When $l\perp x$-axis, $|AB|=\sqrt 2$, and the eccentricity of $C$ is $\dfrac{\sqrt 2}2$.

1. Find the standard equation of the ellipse $C$. 2. Prove that the line $MN$ passes through a fixed point, and find its coordinates. 3. Let $G$ be the intersection of the lines $AE$ and $BD$. If the area of $\triangle GMN$ is $\dfrac 9{20}$, find the equation of the line $AB$.