Other mistake: The security guard clearly throws his gun out of hand even before Catwoman's whip touches it. (00:54:35)
Other mistake: When the Penguin is talking on the staircase to his audience of voters, he gives a badge to one young woman. As he lifts it up it is clearly a pin-on design button, however the Penguin leans forward as presses the badge against her blouse as if it were a stick-on design. The badge then stays on the blouse and doesn't fall. (01:00:30)
Other mistake: At the end as Penguin is dying after coming up out of the water, blue/black "blood" is coming from his mouth. However the cuts and bruises on his face have red blood coming from them, indicating the inconsistency of what color blood he should be bleeding.
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).