Please open the device's system settings, then Accounts and Sync, and make sure that the "master switch" that allows apps to sync in the background in "enabled". This is a check box / switch near the top.
Alternatively, you could open my app's settings, switch to extended mode, then scroll to "System setting" under "Synchronization settings" and change it to the last item, "Sync even when Background data is off".
As for updates, Google Play should notify about new versions, and there is a check box there to "update automatically".