Archon Egg + 3 Red + 2 Yellow = Pristine Egg Ring.

Painted Archon Egg + 3 Blue + 1 Yellow + 1 Purple = Brilliant Egg Ring.

Coloured Archon Egg + 2 Red + 1 Yellow + 1 Green + 1 Blue = Vibrant Egg Ring.

Darkened Archon Egg + 2 Purple + 2 Red + 1 Green = Midnight Egg Ring.

Not sure on chocobo ring….or the hatching pot.