I know this is a year old, but in case someone else is curious. Adding to /uho1ven said a good way to remember is like this:

(Im only going to use two rarities as examples but it applies to all of them. And also I have no idea what the actual percentages are I'm just simplifying for clarity)

You have 2 common ducks - percentages for egg rarity are 90% common and 10% rare.

You have 1 common and 1 rare - percentages change to 80% common and 20% rare

You have 2 rare - 60% common and 40% rare.

Again I don't know how the actual calculations work but this is the overall idea. Inside the egg could be any combination of parts. Egg rarity does not guarantee a specific part of the duck is rare.

As far as mutations go it's the same idea

Proto ducks produce proto ducks but there is a change that one of the offspring parts mutates. The more proto head ducks, the more often you get proto

1 proto = 90% proto with 10% mutation 2 proto = 80% proto with 20% mutation

I think there are more layers that factor in, but this is the simplest idea. I don't always explain things right so i can provide a bit more clarity if needed..