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.

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[![convexity][1]][1]


  [1]: https://i.sstatic.net/wzB0z.jpg