IPhone – What exactly does “erase and sync” mean in iTunes

This Apple Support article provides a quick summary of what syncing does. Basically, as XAlexOwnZX mentions, a sync copies applications, music, and other information between your iTunes account and your iDevice. This process backs up the content (apps, etc.) on your computer.

The advantage of syncing is that if your iPhone gets corrupted, you'll be able to restore it (or its replacement) to working condition, with all of the purchased content that is installed.

The disadvantage is that syncing takes time - however, with iOS 5, you can choose to sync over WiFi and you don't need to tether your iPhone to your computer.

Your list holds the answer, specifically #4. The problem is the fact that when iTunes it prevented from automatically opening when a device is connected, there's a whole list of actions that simply don't happen, sync triggering is one of them.

Re-enable 'Open iTunes when this iPhone/iPad is connected' and your syncs should work normally, and automatically, again.