To remove Outlook (activesync mail) from your inbox.
In your registry editor...
Code:
HKEY_LOCAL_MACHINE\Software\Microsoft\Inbox\Svc\ActiveSync
1. Back it up to a reg file that way you can restore it if it messes anything else up.
2. Delete the key
3. re-open your inbox and the outlook folder should be gone
It may come back again if you make a new ActiveSync or WMDC partnership, but you can always get rid of it again til next time. Syncing normally seems to work fine and doesn't bring it back.