In this regard, are student emails private?
Most colleges and universities offer a . edu email address to their students. Those accounts are NOT private, they are 100% owned by the college. They can do anything they want with such accounts, including delete them.
Subsequently, question is, can I use my college email for anything? Your . edu email address feels like it's yours, but it is property of your university. They can (and will) disable it (permanently, and without notice) in certain conditions. Mitigate the possible damage by not using it for any personal or professional business.
Furthermore, can universities access student emails?
Under the University's Information Technology Acceptable Use Policy, the University maintains the right to access not only employee accounts, but students' accounts as well.
Can universities see deleted emails?
If the system is setup properly, your administrator should be able to recover the email at least if the mail was deleted in last 24 hours, possibly more, but as I said, it depends, so you just need to ask the administrator.