A Right Angle at the Focus

Consider the parabola $C \colon y^2 = 2px$ (with $p > 0$). The line $x - 2y + 1 = 0$ meets $C$ at points $A$ and $B$ with $|AB| = 4\sqrt{15}$.

1. Find $p$. 2. Let $F$ be the focus of $C$, and let $M, N$ be points on $C$ with $\overrightarrow{MF} \cdot \overrightarrow{NF} = 0$. Find the minimum area of triangle $MNF$.