An Incircle Tangent Angle Doubling

The hyperbola $C:\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ ($a,b>0$) has eccentricity $2$ and left and right foci $F_1$, $F_2$. A line $l$ through $F_2$ meets the right branch of $C$ at points $M$ and $N$; when $MN\perp x$-axis, $|MN|=6$.

1. Find the standard equation of $C$. 2. Find the minimum value of $|MF_2|\cdot|NF_2|$. 3. Let $Q$ be a common point of the incircle $P$ of $\triangle F_1MN$ and the hyperbola $C$, and let $A$ be the left vertex of $C$. Prove that $\angle APQ=2\angle F_2PQ$.