Okay this is several questions. For one, what exactly is a Kakuja? I mean, I get that they're extra kagune obtained by cannibalism, but what causes that to happen? How come Kaneki's takes the form of a centipede? All I know is it's somehow related to Yamori, but how did Yamori trigger this change in Kaneki? And why did it cause Rize and Yamori to become one with him?
A kakuja (赫者, red one, kakuja) is a kind of ghoul with a transformed kagune that clads the ghoul's body. Though it is rare, it can occur through repeated cannibalization. When a ward has a higher ghoul to human ratio, cannibalization is more common due to turf wars and limited food supply.
Sometimes, the transformation is incomplete, and such individuals are called half-kakujas (半赫者, han kakuja).
Kagune for kakuja's tend to be more specialized. For example, Arata's kagune was the Armor. It just happened to be that Kaneki's kagune when in his kakuja state take the shape of a centipede.
This could also be associated with Kaneki's mastery at manipulating kagune. In Tokyo Ghoul;Re a member of the Quinx squad named Saiko Yonebayashi has extreme control over her kagune. Because of this, she is able to freely change the shape and size of her kagune, ranging from nets to huge fists.