A Zero-Slope Conclusion

For the ellipse $E \colon \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$), the quadrilateral whose vertices are the foci and the endpoints of the minor axis is a square with side length $2$. A line through the point $(0, t)$ (with $t > \sqrt{2}$) with defined slope meets $E$ at two distinct points $A$ and $B$; the line through $A$ and $C(0, 1)$ meets $E$ again at $D$.

1. Find the equation of the ellipse $E$ and its eccentricity. 2. If the line $BD$ has slope $0$, find $t$.