Revealing mistake: When Batman glides off the rooftop we catch a glimpse of his face as he flies over Gotham. It isn't Michael Keaton, it's a rather crude looking dummy. (01:20:15)
Continuity mistake: When the Penguin is controlling Batman's batmobile, when the batmobile hits a police car, the car flies into a '24 Hours' shop. Between shots the amount of hubcaps and tires hanging on the brick wall around the doorway of the '24 Hours' changes. (01:20:15)
Revealing mistake: When the Penguin is controlling Batman's batmobile, when the batmobile hits a police car a wire attached to the police car, pulling it back is visible. (01:20:20)
Audio problem: On the batmobile screen, the Penguin says "Just relax " but when Batman plays the Penguin's audio back on the tannoy speakers, it is "Hey, just relax " (01:20:30 - 01:26:00)
Continuity mistake: After Batman pulls the wires from the batmobile roof, the wires disappear/reappear during the scene. (01:20:30)
Revealing mistake: When the Penguin drives the Batmobile remotely, the cars it smashes are damaged just before it hits them. (01:20:50)
Continuity mistake: In the scene where the Batmobile diagnostics detects a foreign object, it shows it being located approximately to the front-left of the windshield. It is also a 2D plan view. Yet Batman automatically knows it is BELOW the car and punches through the floor directly beneath him, which is nowhere near where the Batmobile detects it. (01:21:35)
Continuity mistake: As the Penguin is walking down the steps toward the podium, before Batman plays the messages the Penguin said, a man and a woman standing to the left of the podium swap places between shots. (01:23:30)
Continuity mistake: The fruit and vegetables (tomatoes, lettuces, eggs etc.) thrown at Oswald at his speech, litter the stage as well. The amount and position of these items laying all over the stage changes between shots. (01:27:10)
Continuity mistake: When Oswald starts shooting bullets out of his umbrella at his failed mayor speech, the crowd ducks down on the stairs, at the edge of the red stage, laying down their campaign signs. In the next shot some of the crowd and signs have vanished from around the stage, then reappear in the next shot. (01:27:15)
Continuity mistake: When The Penguin starts to shoot at the booing crowd from his umbrella, the same microphone, on the far left, falls from the podium twice in different shots. (01:27:15)
Continuity mistake: When The Penguin shouts "I am not a human being..." he throws his black coat and shirt to the floor. His coat lays in front of him, but in the next shot it is behind him. His shirt also changes position. (01:28:40)
Continuity mistake: When The Penguin is explaining his scheme about stealing the children fron their cribs and sending them to their watery graves, a black line of spit runs down his chin. When he turns around it's gone. (01:29:15)
Continuity mistake: While arranging the plan with the clowns, the Penguin throws his clothes on the floor, which change positions between shots. (01:29:23 - 01:31:00)
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)
Continuity mistake: When the Penguin shoots the clown henchman for questioning his scheme, the clown's white hair cap flies off. In the next shot, as the Penguin kicks the clown into the water, it is back on his head. (01:29:30)
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)
Continuity mistake: When the Penguin is talking to his penguin audience about stealing first, second, third and fourth born children, the penguins chair, box and umbrella stand in the background keep changing position. (01:38:25)
Continuity mistake: Batman is monitoring the progress of the rocket-laden penguins via radar. At every screen refresh, about one per second, the penguins have moved about half a street, way too fast for the way they waddle. (01:42:25)
Revealing mistake: When The Penguin sends his penguin army out into Gotham City, the ripples/waves in the water don't match the position of some of the penguins as they swim towards the exit. (01:43:55)
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).