Two Focal Chords, Two Circles

Through the focus $F$ of the parabola $E \colon x^2 = 2py$ (with $p > 0$), two distinct lines $l_1$ and $l_2$ are drawn with slopes $k_1$ and $k_2$ satisfying $k_1 + k_2 = 2$. Line $l_1$ meets $E$ at points $A, B$ and line $l_2$ meets $E$ at points $C, D$. Let $M$ and $N$ be the centers of the circles with diameters $AB$ and $CD$ respectively, and let $l$ be the line containing the common chord of these two circles.

1. If $k_1 > 0$ and $k_2 > 0$, prove that $\overrightarrow{FM} \cdot \overrightarrow{FN} < 2p^2$. 2. If the minimum distance from $M$ to the line $l$ is $\dfrac{7\sqrt{5}}{5}$, find the equation of the parabola $E$.