Outlook 2011 keeps everything, mail cache, sync state and settings, inside an identity, a database in your user Library. That database corrupting is the overwhelming cause of exactly these symptoms. The server being current in webmail confirms it: the identity's record of what has synced no longer matches reality and the client cannot reconcile.
1
Rebuild the identity with the bundled tool: quit Outlook, hold the Option key and launch Outlook again, which opens the Microsoft Database Utility instead.
2
Select your identity and click Rebuild. The utility copies the identity aside and reconstructs the database, taking a while proportional to mailbox size, leave it alone until done.
3
Launch Outlook normally afterwards. With Exchange the client now re-verifies folders against the server and the stuck states clear as the fresh database populates.
4
If rebuilding fails or the problems return within days, retire the identity instead: create a new one in the same utility, set it as default and add the Exchange account fresh. Since Exchange holds the mail server side, the new identity downloads a clean copy and nothing is lost.
Check the free space on the Mac before either step, identities double in size temporarily during a rebuild and a nearly full disk is itself a common trigger of the original corruption. And the era appropriate caveat applies here as elsewhere: 2011 is unsupported and its EWS dialect ages against modern Exchange, so recurring sync trouble on a healthy identity is often the server outgrowing the client rather than anything fixable locally.