Circle on Diameter OM Meeting an Ellipse

The ellipse $C_1: \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ $(a > b > 0)$ has eccentricity $\dfrac{\sqrt{2}}{2}$, and the lower endpoint of its minor axis lies on the directrix of the parabola $x^2 = 4y$. Let $O$ be the origin, let $M$ be a moving point on the line $l: x = 2$, and let $F$ be the right focus of the ellipse. The line through $F$ perpendicular to $OM$ meets the circle $C_2$ with diameter $OM$ at two points $P$ and $Q$, and meets the ellipse $C_1$ at two points $A$ and $B$, as shown in the figure.

(1) Find the equation of the ellipse $C_1$.

(2) If $|PQ| = \sqrt{6}$, find the equation of the circle $C_2$.

(3) Let $S_1$ and $S_2$ denote the areas of the circle $C_2$ and of the quadrilateral $OAMB$ respectively. If $S_1 = \lambda S_2$, find the range of $\lambda$.