Going back to breeding, I set up a Cattery trying to breed them quickly.
I have 2 cats per room with 2 bowls 2 beds and the haystack. I have assigned the names to the room. Got them to 100% and fed them the special strawberry.
Still no Kittens. My cows seem to be breed non stop though.
Am I missing a step.
First, what language are you playing in? Apparently there are Male and Female cats (though you can't see that in English) and you need to breed a male and a female together to get a kitten, supposedly. You can see that in languages with gendered nouns, like Spanish. FWIW, I never had any problems breeding both types of pets all the way to the final coloration in English despite not being able to see the genders of pets.
Secondly, only breed one pair per day. You may be running into a 牧場物語 (Story of Seasons) issue where you can only have one "event" per morning, which also includes birth events. Cows, Chickens, and Sheep will breed until there are 5 in any given area, so make sure you have 5 to stop their breeding, and/or send them to another area.
Third, and most importantly, do NOT hang out near the area the pets are in. If you must be in that area, hang out near the pet houses when it is sleeping time. The Builder can kind of distract them from going to sleep properly, and sleeping in the great outdoors will lower the pet happiness enough that they won't breed that day.
As another note, you don't need 4 Gen 1 parents. You can use the same pair to make two babies on consecutive days, then send the original pair away and raise Gen 2 to adulthood and continue breeding pairs with the Gen 2 pets. Lather, rinse, repeat. The game doesn't really have a problem with inbreeding and it is the most efficient way to get to the Gen 4 colorations.