will carrier settings update with the apple update

Is your phone carrier locked to AT&T? Did you ask AT&T to unlock your phone? Only the carrier a phone is locked to, can unlock it. You need to contact AT&T to see if your phone is locked, and what their requirement is for you to move to another carrier. That would be between you and AT&T and Apple has nothing to do with that.

No, it will be a problem that your iPhone 7 is not registered in Verizon’s system. I’m not sure if they will still accept this phone, call their support number and if necessary ask for tier 2 support.

Yes, if your cell provider has issued a carrier update.