Three Lines Through $P$

Consider the parabola $C \colon y^2 = 2px$ (with $p > 0$). Through the point $P(2, 1)$, lines $l_1$ and $l_2$ with slopes $k_1$ and $k_2$ meet the parabola at $A, B$ and $M, N$ respectively. When $k_1 = 2$, $P$ is the midpoint of $AB$.

1. Find the equation of the parabola $C$. 2. If $|PM| \cdot |PN| = |PA| \cdot |PB|$, prove that $k_1 + k_2 = 0$. 3. If the line $AM$ passes through $Q(-2, 0)$, prove that the line $BM$ passes through a fixed point, and find its coordinates.