While it might be theoretically possible to charge everyone in a car with the possession of the alcohol container, I have found most prosecutors (and probably most courts) do not like that shotgun approach. To be more certain of a filing, you narrow it down to the person who had immediate access - in this case, that is apparently the OP's daughter. And, it might be that the daughter even indicated she was aware of the jar and what it contained, even though it was not hers. People sometime forget that "possession" does NOT mean "ownership". Itmeans only that constructive possession (dominion and control) of the item in question.
The "empty" beer cans in the back cannot be charged as an open container because, well, they are empty. Unless, of course, the law in MI covers the possession of empty beer cans (which might make dumpster diving and recycling a legally vexing occupation).
I agree that the daughter needs to look into the possibility of consulting an attorney. There may be available diversion programs or other options to help keep this off of any permanent record.
As for notifying her school, I don't know that would happen. The state is under no obligation to seek out a defendant's employment or school info (and we never sought it for notification purposes), and a defendant is under no legal obligation in MI (that I know of) to supply this info to the police. I DO know that in my state the police do not regularly make such notifications, and it is arguable that such a contact for no legitimate enforcement purpose could be seen as some form of unlawful harassment ... the term escapes me as it has been a long night. A University officer, on the other hand, likely could potentially report the offense to the university as a matter of course.