You can select all the projection objects and target surfaces before starting the command.
The curves are projected vertical to the construction plane that is active when the surface selection is completed.
If the projection misses the selected surfaces and polysurfaces, a curve will not be created. Make sure the correct construction plane is active when you select the surfaces.
The Pull command will suck the curve back toward the surface by closest points. The Project command will not work in situations where you want to pull a curve onto a cylinder when the curve goes most of the way around the cylinder. Use the Pull command in this case.
The Project command creates complex curves that can be simplified with the Rebuild command. You will need to be careful with the Rebuild command and use enough points to keep the curve trimmable.