So far, all of my (few) graduated idol are staying in the salary tab and get payed, they are also still eligible for events and preset formation.
They also stay in media projects (the stamina cost doesn't change for the other idols in the same project).
Graduated idols don't count against the 5 idols requirement for elections, and don't show up during the election, but I was able to select them when I wanted to swap the envelope.
I tried all the three options (staff, successor and goodbye), and made sure that they aren't in any media projects. I also restarted the game for every try, to make sure there is no weird interference, and waited ingame for at least a month after the graduation.
I also tried to fire them by myself. Fire them directly only works after the second try on the same girl. It has the same salary/event problem like the normal graduated idols.
It also seems you get a lot more freezes the more graduated you have. During my short testing, I tried to run a save with 3 graduated idols and one soon to be. The game freezes all couple days for no obvious reasons. Restarting the game did unfreez the game for a couple days.