Adding to #1, it'd be nice if graduated idols were moved to some sort of "hall of fame" where you could view their profiles, or something along these lines. As a bonus, this could open up a possibility to (still) interact with them in some manner.
Also, maybe an option to set a concert as a "graduation ceremony" for an idol that's about to leave soon? It could make the fan base of that particular idol more likely to attend, or smth.