I believe this answers (1). $P$ is the pyramid illustrated. $S$ is a square resting on the apex of $P$, at height $z_1$. Projecting $S$ down (green lines) onto $P$ results in the nonconvex shape outlined in red. The projection only reaches $z_2$ on the thin side faces, but much further down to $z_3$ on the front and back faces. The front/back faces are slanted more steeply than the left/right faces. So, Yes: Projection of a convex set $S$ onto a convex set $P$ can be nonconvex and connected.
