There are gold-eyed bastards up and down the eastern coast of the Earth Kingdom, and more than a handful in the Fire Nation. However, he cannot publicly acknowledge them without removing his two legitimate boys from inheriting the throne. Fire Nation has very strict primogeniture, the way I write it, and his bastards tend to be either older than Iroh or younger than Ozai. (He didn't stray much while his wife was alive.)

I suspect Sozin invented ragebending and/or it was a minor style of firebending before his time. Azulon, I get the impression, did a lot of polishing on it.