I understand copyright happens at the point of creation, but if I Iunderstand © law correctly, to claim financial damages, you need to register with the copyright office. Most online creators/bloggers don't register. Can they sue for more than just removal of their works from AI and be awarded $?

You can register after the fact but you need to register before you can sue for damages. And that's not touching on the cost of litigation. Registration is trivial compared to enforcing it.