Here is the save file location. Because he had 2 options,'enroll university' and 'apply university(75)'. At first time, I didn't use apply university and tried to enroll the university, and all distinguished degrees were locked except for those 2 degrees,'art history and fine art' in britechester. So I expected the locked degrees would be activated if I apply the university one more time, and I applied and got the acceptance letter from the university. That's why apply university is grayed out and only enroll the university is left activated.
Yes Once the sim graduated the university, that sim doesn't need to apply the university again when that sim wants to challenge another one. In case my G sim, he keeps enroll the university and can start a new term straightaway without re applying university.