I'm a NE admin, so I can't spare much time looking into these possibilities, but I can give you a couple places to look.
For Outlook sync, there are a number of projects/products out there that
purport to offer integration. On the server side if you could somehow plug
OpenChange into Zimbra, you might have something. (Looks like SOGo is already headed in that direction.) On the client side there's
The Open Outlook Connector Project and ZideOne (may not be open source). The former looks like development has stalled; the latter seems to have been abandoned completely.
While doing more search I came across
iCal4OL: Google Calendar Synchronization for Outlook - ICAL Feeds Import and Export but that is commercial only.
As for mobile sync, search the forum for funambol and z-push. Not sure about document sharing, in fact, are you sure it's a NE-only feature?