Evolution can be nigh-impossible to understand sometimes, but this one seems straightforward- as you mentioned, it was originally two discs, which gained bumps- bugs that had better bumps were more likely to survive, and thus the bumps probably became more defined and streamlined, forming ridges, and later "gears"- and if there's one thing the body likes to do, it's symmetry.

at the end of your last sentence, I'd generalize that to "if there's one thing the universe likes, it's symmetry."

repeating patterns are common due to segmentation and gene duplication. I suspect, and I think a lot of others do as well, that the appearance of complicated new shapes occurs fairly commonly after mutation, as a consequence of the hierarchical structure of developmental gene regulatory networks and the process of cellular differentiation. In a sense, it could be there is a collection of "modules", and successful mutations involve mostly manipulation of the regulatory network of those modules, while the modules themselves don't change very often at all (at least in part because they have so many dependent processes). By combining expression of those modules with spatial patterning.

Well, a long series of repeating patterns happen very often in organisms and have some well understood evolutionary paths.

It was even Turing that discovered how some of those patterns form.