#1 would depend on how you want to pause it. Do you want the entire game to stop? If so you need to set Time.timeScale to 0. If you only want the ship to stop, then your best bet is to pause the sampling of the spline. The ship there simply follows a fixed spline by sampling it in update.
Same with #2 -- since you are the one who's sampling the spline, you can simply sample it ahead of where the ship would normally be, and temporarily move the ship there.