Revealing mistake: When The Penguin shoots the clown who says "Killing sleeping children...", as the clown falls back to the floor, you can clearly see he falls on a wrinkled, soft mat which has unsuccessfully tried to be disguised. (01:29:30)
Visible crew/equipment: When the Penguin dies and the penguins slide him into the water you can see the slide-board he was laying on slip out ahead of him under the water. It is also very clear that the penguins aren't even touching him as they pull him into the water. (01:53:00)
Continuity mistake: When Catwoman exits Schreck's she's doing flips without hells. After the explosion, when she climbs the wall heels appear.
Continuity mistake: After The Penguin dies Batman's car drives next to a white car with its windshield covered in snow. From the shot inside Batman's car the snow is gone.
Continuity mistake: Inside the cage, Max covers his mouth with his hand and then uncovers it and grabs an iron bar. A shot later he is suddenly covering his mouth again. (01:37:16)
Answer: The film keeps it ambiguous. There are two ways to look at it. Scenario A- She literally died and is brought back to life by the alley-cats somehow, adapting some of their traits. Or... Scenario B- The trauma of falling and sustaining a head-injury, along with her paranoia after Schreck tried to kill her, has driven her mad, and she uses her connection with cats to build a new persona. (Which is supported by the fact that all the things she claims "kills" her with each of her "nine lives" wouldn't actually have killed her. Ex. Her one fall is broken by the kitty-litter truck. Max doesn't hit her in any vital organs when he shoots her. Etc.) Her destroying her apartment is her lashing out at all the things she used to hold dear- her vision of a normal life, etc. It's symbolic of her purging the past and embracing the future. (Plus, oftentimes when people throw fits, they'll smash stuff up).