The cat is not getting enough exercise/stimulation. Cats like to hunt and pounce and if they don't have anything for them to release that energy on then they'll **** shit up. 'til the cat mellows out the dog will probably be hated.
Oh and if you start being it's "provider" by feeding it expect it to come to you whenever it thinks it's hungry, even if it has food or it's the middle of the night.
Why not keep it in the pet carrier/kennel part of the time?
The left wing and the right wing belong to the same bird.