Since you get your music from a legal music service run by the school, there should be no issue obtaining music you are entitled to anyway. The lady who got expelled for downloading through P2P was most likely caught with the illegal tracks either on her account's storage space or a technician seen the file sharing activity on the Internet logs for her. In the case with the lady, she was not only putting herself at risk from an RIAA lawsuit, but also potentially putting a bad name on the school itself for allowing P2P file sharing. I'm sure there is also a policy in place where a student can face being expelled for using the school's Internet connection for copyright infringement.
As far as I'm aware of, there should be no issue using Tunebite to convert the tracks to make them compatible with your iPod, so long as you don't make CD copies of them (as stated in the agreement) or leave any of the converted files on your account's storage space or any of the PCs. Since you're entitled to listen to the music as long as you're a student and you are not sharing out the songs (as the expelled lady likely ended up doing with the use of P2P), I cannot see anything wrong with making them compatible with your iPod. However, as the provider only allows the music to be playable as long as you're a student at the school, by right you would need to delete all the music you have obtained and converted from the school once you finish up with your course.