The updates that made this issue worse are: KB2837618 or KB2837643, so please avoid running this update until Microsoft corrects the issue. If you cannot avoid the update, please see below on how to solve it:
IMAP emails do not show up in Outlook 2013 even though all settings are correct and you have no errors:
ok this one seems to actually be a pretty simple fix:
Click on Folder (the inbox folder)>AutoArchiveSettings>AutoArchive> Then Check the box which says 'Do not archive items in this folder".
UNINSTALL the updates:
Open the Start menu and click visit your Control Panel.
Click Programs and Features
Click View Installed Updates
Locate either the KB2837618 or KB2837643 updates and remove them.
you can also try this solution:
Under the "Folder" tab, right of the "Favorites" tab, click "IMAP Folders"
Outlook should open a dialog box with IMAP folders.
Remove the check from the box that says, "When displaying hierarchy in Outlook, show only the subscribed folders."
Force sending/receiving within Outlook and everything should now synchronize.
Emails and profiles will not delete (you get an error message saying can not delete as its already deleted):
see this help article on how to repair your PST file