In line with my comment below, what do you do when $S$ is mapped onto the origin? (This happens when $S$ is wholly contained in $-C$.)

It is not always the case that a ray is projected to a ray (in the harder question): for example, the projection of the ray $\mathbb R \cdot (-1,-1)$ onto the positive quadrant $\{(x,y) \mid x,y\ge 0\}$ is just the origin.