CTRL+ RMB (Left Click Select) or CTRL+LMB (Right Click Select to draw points and SHIFT on a curve point to pull out handles.
Enter to sculpt the curve:
Is it possible to stick the curve to the mesh (instead of the view) ?
As far as I know, not...that would defeat the purpose imho.
The curve can be used on different parts of the Mesh, with different Scale, rotation and even on different Objects.