If he received any student aid, loans, scholarships, he would have had to register. You have so show proof of SS registration in order to receive these.
That is correct. Excellent point.
I think this is an interesting question, and should probably be followed up. My guess is that there is a good chance he didn't register. I remember those days, and given his political tendencies, he may have protested by not registering.
FYI, Obama did not move to Chicago until 1985. In 1980, he would have been either in Los Angeles, where he was attending Occidental College, or back home in Hawaii.