If I sync my iPad that has apps put on there by my school, will they be erased if I sync only music?
No. The moment you plug it into your home machine, it will ask you to format it as a new iPad. It is tied to one account, one iTunes.
While I know this is true, I knew there was a utility that used to be there in Cydia to sync across various iTunes accounts. Any idea on what it was? At the time, I had not needed it and now I cannot seem to find it!
It hasn't done that because I was able to manually move files onto it. I can't jailbreak it either.
You can also transfer data to and from your iPad if your school allows installation of software like Copytrans manager -
Make sure "Manually manage music and videos" is checked, plug-in youipad and click on the option that says check items you want to sync