Hi there. First of all, there is Course completion and Activity completion (The links go to the documentation)
Activity completion is set for activities and as a teacher in the course you will get a link in Administration> Course administration> Reports > Activity completion to see who has and has not completed the work. See [[Using Activity completion]
With respect to Course completion just to point out that it is not currently possible to set a course up so that a user cannot access another course until they have completed the first one something you alluded to in your post. The Course completion status block enables teachers and students to monitor their progress towards completion of a course
I can't help with the notifications issue sorry but hopefully others with experience of notifications can.