Goldfish need to be about 2 years old to breed.
They're also egg scatterers, meaning they scatter the eggs all around the bottom of the tank.
Without knowing any more info, it's possible that's what they're doing. It's also possible the one is just bullying the other.
Small white cotton thing, sounds to me like something is rotting and turning into mold in the tank. Could be eggs, but like I said, they'll be all over the bottom, not just one area. And if those are eggs, if you see fungus on them, they're bad and won't hatch.
