Folding a Right Trapezoid into a Hexahedron

In the right trapezoid $ABCD$, $BC\parallel AD$, $AB\perp AD$, $BC=8$, $AD=9$, and $AB=2\sqrt{3}$. Point $E$ is an interior point of segment $BC$ (not an endpoint), and the line through $E$ parallel to $AB$ meets $AD$ at $F$. The rectangle $ABEF$ is then folded up along $EF$ until its plane is perpendicular to the plane of the trapezoid $ECDF$, producing the hexahedron $ABCDEF$.

1. If $CF\perp BD$, find the length of $BE$.

2. Find the minimum value of the cosine of the angle between the skew lines $BC$ and $AD$.