Do I have to register a copyright?

No. Registration is not required, it is optional. But it is required for a court action of infringement.

“..registration is not a condition of copyright protection.”
— U.S. Copyright Office, Circular 1

“No publication or registration or other action in the Copyright Office is required to secure copyright. There are, however, certain definite advantages to registration.”
— U.S. Copyright Office, Circular 1