You must NEVER, really NEVER apps, which are installed by the system installer. The AB database is not in the app itself though, therefore deleting the app does not delete the database. It is located at the user level in ~/Library/Application Support/Address Book. If you are in 10.5.8 as shown, you may see it there.
How did your reinstall the AB? There is no installer for it, it is within the system installer.
Your second problem is syncing, which is indeed connect to AB as well. Why did you delete the app, when you knwo it requires syncing with your iPhone?
Your third problem is that you do not have, I assume, regular backup of your data. If you had this, you would not have such problems. So choose a backup utility at your choice. If with system 10.5.8, I think you may still use Apple’s Backup, if not you may choose a 3rd party backup utility, there are several, check with macupdate.com. I may recommend iBackup of grapefruit.ch, but there are others too.