Revealing mistake: When The Penguin controls the Batmobile, before it crashes against a car on the left and sends it flying away, check the car's rear wheels: they're not touching the road.
Revealing mistake: There is a shot of Batman driving the Batmobile around town late at night looking for The Penguin. In this shot, look at the cars parked at the side of the road. They are miniatures as well as the rest of the shot.
Revealing mistake: When the penguins emerge to carry the Penguin (character) to the water after he dies, the penguins are blatantly little people in costumes. Their "skin" crinckles unnaturally, and the spot where their arms meet their bodies is very flat and fake-looking.
Revealing mistake: When Batman is driving the Bat-boat, just after he dodges the two rockets the penguins launch at him, as the Bat-boat slams back down into the water in the background, the scale of the water-splash reveal that this effect was achieved using a miniature set and miniature Bat-boat model. The scale of the water splash, and the enormous "drops" that splash up give the effect away.
Revealing mistake: When The Penguin (character) is first introduced, feeding the other (real) penguins, watch closely. While some of the penguins are real, it's blatantly obvious that some of them (particularly the ones to the left of the screen when Penguin says "Hi") are animatronic puppets, as they have very unnatural neck movements and don't waddle around like the rest. A few shots later, when Penguin says, "I believe the word you're looking for is...", you can even see an animatronic penguin on the left side of the screen that is unnaturally leaning slightly forward, yet never falls, revealing it's a puppet that was mounted to the ground.
Revealing mistake: When the penguins launch the rockets/missiles in the end, watch carefully. In the second wideshot of the rockets launching (right after the bats begin to attack Danny DeVito), you can see two of the rockets hanging in midair, not moving. At the beginning of the shot, you can even see one of them spurting some fire. The rockets (attached to visible guided wires by the effects crew) got stuck in mid air right before the shot began, a few seconds before this second volley of rockets fire off.
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).