A Circle Tangent to the Y-Axis

The ellipse $E:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a>b>0$) has upper vertex $D(0,3)$, and the quadrilateral formed by its four vertices has area $18\sqrt 2$.

1. Find the equation of the ellipse $E$. 2. A line through the point $T(0,1)$ meets $E$ at two points $A$, $B$ and meets the $x$-axis at a point $Q$. The lines $DA$ and $DB$ meet the line $y=t$ at points $M$ and $N$ respectively, and $P$ is the midpoint of $MN$. Does there exist a real number $t$ such that the circle with diameter $PQ$ is always tangent to the $y$-axis? If so, find $t$; if not, explain why.