A Slope-Product Fixed Point

The ellipse $E \colon \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$) has major axis of length $4$, and $D$ is its right vertex. For any four non-vertex points $P, Q, M, N$ on $E$: when the line $PQ$ passes through the origin $O$, the slopes of $PD$ and $QD$ multiply to $-\dfrac{1}{4}$.

1. Find the equation of the ellipse $E$. 2. When the slopes of $MD$ and $ND$ multiply to the constant $-2$, does the line $MN$ pass through a fixed point? If so, find it; if not, explain why.