Cats can learn to open doors and they are so fast and quiet (def would get those squeeze knob covers for the handles that they use around children to keep doors shut). I wouldn't have a cat and bird...ever....I know people do it, but they carry bacteria that can kill a bird from a small scratch.

Seems like the birds will have to be shut away all day alone so that the cat can roam free..

Do you have to get the kitten? lol

Even if a cat seems to ignore a bird, it only takes one moment for things to go wrong, and they are programmed to hunt (hense all of those fishing pole+ feather on a strong toys...chasing lazer pointers etc)...and they have small claws that can fit in small areas...

I just wouldn't be comfortable risking it. They are far to efficient at killing birds...It is a personal choice, but not one I would ever make.