> Doesn't Amazon shipping have to go to the billing address on the credit card?

No and that would be crazy. I'm not aware of any e-commerce site that has a restriction like that.

> Being able to purchase on a credit card and have it sent anywhere makes it that much easier to use stolen credit cards.

Well, it's probably one fraud signal among many, but it's absolutely not generally prohibited. I've sent things from Amazon to other people (or to myself while staying in a hotel), and other people have sent things to me, many times.