Hello Lisa & welcome!

I can't really comment on the cats, although I have been told most cockapoos seem to like cats (although the feeling isn't always reciprocated lol)
Size can vary depending on the breed cross (toy/miniature poodle, american/english cocker etc). American cocker spaniels are smaller than English, & obviously toy poodles are smaller than miniature. What kind of size are you wanting?
My Izzie is an English show cocker spaniel x miniature poodle as she is about 14.5" to her shoulders fully grown (so quite small), but it always helps to see the parents & see how big they are as sizes can vary in breeds, Izzie poodle dad isn't a big one, & she is exactly like him looks & colour & size wise

There aren't small children in my house either, but I have a cousin who has a little girl & Izzie has always been great with her

So I would say they are good with children from my experiences.
Although obviously you would need ot make sure that the breeder you choose is breeding for temperment

So again it helps to meet the parents so that you can see what their personalities are like

Good luck with your search! There are always breeders you can ring up to ask your questions? Just as Anzil cockapoos & Jukes Doodles, they will help with any advice from a breeders perspective
