Hi Peer, yes like Pilou says Construct > Curve > Project or Construct > Curve > Intersect.

So for projection you would set up lines like this:



Select them, then run Construct > Curve > Project and select the object as the projection target:



re:
> hoping that I could snap to the intersection between the cylinder axis and the surface of the sphere, but snapping to that point didn't pan out.

Surface intersection can be an expensive operation so those snap points are not automatically calculated. But you can generate a point
object that you can snap on to between a curve and a surface by selecting them and running Construct > Curve > Intersect.

- Michael

Michael and Pilou,

Thank you both for helping me! This method worked perfectly.

And thank you Michael for telling me about construct > curve > intersect. That's good to know.