Outlook is rather sub-standard in that it does not notify you when the server tells it that you are running out of space. The server is configured to notify you when you go over 80%.

You can check your quota, using the websieve tool Can I set up Automatic Replies and Vacation Messages?

Why does my deleted mail stay in my inbox rather than going into my deleted folder

If you're a pop3 user, you've probably set up (or not unset) your POP client to leave a copy of your mail on the server, and this is a different symptom from The same messages appear again as new every time I log in with pop but you should again unset the flag to download and delete your incoming mail. (details in the link)

Normally POP3 mailboxes do not run into this problem