Since Google seems to be working (including gmail and Youtube), root has nothing to do with contacts (or Google) and unlocking has nothing to do with the computer part of the phone, I can't come up with a guess to what's wrong. Unless someone else has an idea, I suggest you restore stock 4.3, then see if you still have a problem.
If you do, you can bring it to the carrier to ask for help. If not, root and see what happens. (Since Gmail, Youtube and Contacts aren't rooted programs, they shouldn't work any differently after you root.) Add your apps, one by one (you do have them all backed up off-phone, right?) and see when you start having a problem. That's the app causing it.
If you restore them all and there's no problem, unlock and you're good. (Unlocking has nothing to do with the computer part of the phone or anything Android.)
10-24-2014 10:34 PM