A Rhombus on a Focal Chord

In the coordinate plane, the ellipse $C_1 \colon \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$) has foci $F_1, F_2$, where $F_2$ is also the focus of the parabola $C_2 \colon y^2 = 4x$. The curves meet in the first quadrant at a point $P$ with $|PF_2| = \dfrac{5}{3}$.

1. Find the equation of the ellipse. 2. A line through $F_2$, not perpendicular to either axis, meets the ellipse at $M$ and $N$. Suppose a point $T(t, 0)$ on segment $OF_2$ is such that the parallelogram with adjacent sides $TM, TN$ is a rhombus. Find the range of possible values of $t$.