A Centroid-Locked Chord

The parabola $C \colon y^2 = 2px$ ($p > 0$) has focus $F$, and the point $P(4, t)$ on $C$ (with $t > 0$) is at distance $5$ from $F$.

1. Find $p$ and $t$. 2. Let $A, B$ be points of $C$ such that the centroid of $\triangle PAB$ lies on the line $y = -\dfrac{4}{3}$.

1. Prove that the slope of line $AB$ is constant.
2. Let $Q$ be the intersection of line $AB$ with the $x$-axis, $T$ the midpoint of $AB$, and $R$ the midpoint of $PQ$. The perpendicular from $P$ to line $TR$ has foot $H$. Prove that $H$ moves on a fixed circle.