I added a payment method to download free applications, and apple charged me $13.00 US. I was charged $13.00 for absolutely nothing.

This is primarily a user-to-user support forum; other Apple users like you. We don't have access to information about charges to your account. See each of the steps in the following instructions. They include information on contacting Apple Support if the other steps don't help: If you see 'apple.com/bill' on your billing statement - Apple Support

There is not much we can do about that, but to tell you to set up a online call or go to a Apple store and have them help you.