Do I have to register with your office to be protected?
No. In general, registration is voluntary. Copyright exists from the moment the work is created. You will have to register, however, if you wish to bring a lawsuit for infringement of a U.S. work.
They can't bring a lawsuit against us because Danbooru is US based and they certainly don't have a registered copyright for Big Tiddy Anime Waifu Pic #296468.
In addition, fanwork is not protected. Further, anything listed as a commission or similar cannot be claimed under copyright by the artist. Works for Hire are under copyright of the person who commissioned it.
As such, a vast majority of images on the site are not protected under copyright, even with a repost notice.
Lastly, as Danbooru is non-commercial and public display, anything posted on public media (Which includes pixiv, twitter, etc.), by both American and Japanese copyright statutes, is fair game to be publically reproduced (again, which Danbooru is) for non-commercial, non-transformative use.
It would be nigh impossible for a Japanese artist to take Danbooru to court over reposting a picture of Venti's butthole or whatever.