All mailboxes you see under your e-mail account are a part of the e-mail account and are saved both on the computer and the e-mail server. Since the account is configured as IMAP, any change you do on your computer, such as delete an e-mail, move an e-mail, etc, will automatically be synced with the e-mail server.
However, all mailboxes you see under the Storage folders are not a part of the e-mail account. The mailboxes there represent mailboxes that are saved only on your computer, and not on the e-mail server.
So, if the Trash is under Inbox, it means it a part of the e-mail account. It probably cannot be deleted or renamed since it is used by the e-mail server, but you should be able to choose another mailbox to put the deleted e-mails into.
Since you say the e-mails now don't go anywhere, please do this test:
1. Right-click on the e-mail account and select New folder...
2. Give it any name that is not taken yet, for example, test123, and click OK
3. You should then see test123 created under your e-mail account, not under Storage folders
4. Now right-click on the e-mail account again and select Properties
5. Click on IMAP
6. For "Deleted Items path", type test123 and click OK
7. Click Yes to refresh the folder list
8. Now delete an e-mail message. Check if it went to test123 or elsewhere