I've not had the music get stuck like you describe to be of help with it. However, the psychic help opportunity at the police station has never worked for me. I researched it before and found that it was glitched with no solutions offered. I periodically try it when there's been a patch or when it comes up for a sim with a different career than before, but it always behaves the same (can go and do it but it never completes). I always just refuse the opportunity now or cancel it if I accidentally accept before realizing what it is.
When I've had the study at the library opportunity come up, I've been able to click on the sim while at the library, and study is one of the options. If it's not showing up, you might click on the phone or a computer while at the library and study that way. I don't remember doing it that way, but it might work.