requirements

You don't need to have any documents to purchase a new phone if you are paying cash or with a credit card.

If you are paying on a loan program or via the iPhone Upgrade Program - Apple you will need proof of identity and a credit check will likely be run.

To trade in your phone, read the page here:

Apple Trade In - Apple

You don't need documents but the phone has to be fully paid for and owned by you, it can't be on any carrier payment plan.

Yes, they will run a credit check if you are purchasing a new phone through the iPhone Upgrade Program.

A carrier will run a credit check if you are purchasing a phone on a monthly payment plan with them.

Generally the checks don't require documents other than identification and other information required when opening a credit card.