Hi,
Concerning syncing an Android phone and Zimbra, we developped a little application called "LVContacts Sync for Zimbra" which enables address books syncing between Android and Zimbra.
It lets you sync multiple address books (not only the default one) and even shared address books.
The synced address books are then tightly integrated in the Android's address book so that you can use those contacts as if you created them with your phone (so you can pass a call, send a message, send emails directly, no need for a dedicated application).
Concerning calendars, we're also working on the same kind of application with the same level of integration but it is not yet available (expect it to be released in September) ... at that moment, you can have a workaround using the application called "Calendar Sync" but it works for the default calendar only, you won't be able to sync more calendars.
Concerning mail syncing, we're planning this too but no scheduled release by now, as a workaround, you can use IMAP sync with the mail application, it works good but there are some little problems with the special folders which are not the same between the application and Zimbra so it creates duplicated folders ... niot very annoying anyway.
So, concerning our application "LVContacts Sync for Zimbra", it is available through the Market (try to type "zimbra" in the search box), it need Android 2.0 as a minimum but it's not validated on 2.2 at that moment (but we're working on it) AND ... it works with any release of Zimbra since it doesn't need ActiveSync, it uses the native Zimbra SOAP API and thus, it works with Zimbra NE, OSE and even some Zimbra for ISP.
You can get more informations/documentations here :
LVContacts Sync for Zimbra and Google Android
Regards.