Can I "speak IMAP" and review my actual mail quota?

Perhaps the Python imaplib library could help. Launch a python console from the terminal with python. Then, create an IMAP connection with the following commands:

>>> import imaplib
>>> conn = imaplib.IMAP4('hostname')
>>> conn.login('username', 'password')

If your IMAP server uses SSL, use the IMAP4_SSL constructor instead of IMAP4. You can then use the getquotaroot or getquota methods on the connection. For example:

>>> conn.getquotaroot('INBOX')
>>> conn.getquota('quota root') # using the root from the previous command