Do ellipsoids cast ellipsoidal shadows?

Yes they do. You can prove it by induction on the codimension of the subspace you project to. For $x\in Vect(e_1,\ldots e_{n-1})$ there exists $t \in \mathbb{R}$ such that $x+te_n$ belongs to $\Delta$ iff the discriminant of the degree $2$ equation $(x+te_n)^TA(x+te_n)\leq c$ w.r.t. the unknown $t$ is non-negative, which turns out to still be a quadratic inequality in $x$.

Yes. An ellipsoid is a linear transformation of a spherical ball, and orthogonal projection is also a linear transformation, so it suffices to show that any linear transformation whose image is a subspace sends a spherical ball to an ellipsoid in that space.

A linear transformation can be decomposed into orthogonal projection by its kernel followed by some invertible linear transformation. Orthogonal projection sends a spherical ball to a spherical ball in the subspace, so we are done.

Indeed, ellipsoids cast ellipse shape shadows on the ground.

The intersection of any conicoid and a first degree equation plane illumination terminator between two tangential points is a conic section. It can be proved by elimination to the conic second degree equation.