A Hyperbola Checklist

The line $y = kx$ meets the hyperbola $\dfrac{x^2}{4} - \dfrac{y^2}{3} = 1$ at points $P$ and $Q$, with $P$ in the first quadrant. Let $N$ be the foot of the perpendicular from $P$ to the $x$-axis, and let $F$ be the left focus. Determine, with proof, which of the following are true:

1. if $|PQ| = 2\sqrt{7}$, then $PF \perp QF$; 2. if $PF \perp QF$, then the area of $\triangle PQF$ is $4$; 3. $\dfrac{|PF|}{|PN|} > 2$; 4. the minimum value of $|PF| - |PN|$ is $4$.