There should be (a sync all button)....
Anyway,i configured zimbra to refresh all accounts every 1 minute - the minimum time setting - to "overcome" a little bit the abscense of the global synchronization. If you have many accounts to check,you can check one or two important ones and let Zimbra check in the time of the 1 minute all the others. A "semiglobalsync" function.
But,still, a global sync feature via button must be implemented...
Contacts get deleted with any restart of Zimbra. The remain saved only for the current run session of Zimbra. With any shutdown and restart,they are gone.
The accounts are 3 gmail and 1 yahoo.