Two Lines Meet a Vertical

The ellipse $C: \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ ($a > b > 0$) has major axis of length $4$, and the vertical chord through its left focus has length $1$.

1. Find the equation of $C$. 2. Let $A$ be the right vertex of $C$. A line through $B(1, 0)$ meets $C$ at points $E$ and $F$; the lines $AE$ and $AF$ meet the line $x = 3$ at two distinct points $M$ and $N$. Find the range of $\overrightarrow{EM} \cdot \overrightarrow{FN}$.