I would like to know if it is possible to pull surface control point on to a mesh. I would like to create a surface using curves projected on a mesh, but I would like to control to snap to the mesh to give me an accurate final surface.
You can snap a control point to a vertex, but it will not be accurate. Projected curves will get you closer to original mesh.
It will not - surface control points do not necessarily lie on the surface itself, so snapping them to a mesh does not guarantee that the surface will coincide with the mesh at that point - in fact it’s almost guaranteed not to.