Why did Batman fall for Catwoman in DKR?

I recently watched The Dark Knight Rises for a third time in theatres, and a question kept popping into my mind (one of many, actually).

Why did Batman care about Catwoman AT ALL?

Why did he choose her as his future, beyond the cape? There are a lot of head-scratchers in that movie, but that one is right at the top of the list. So let’s review! What did he see in her? To get to the bottom of it, let’s look at what, exactly, she does during the course of the film to make him fall for her.

1. She steals his dead mom’s pearl necklace, as well as Bruce’s fingerprints.

2. Admits she robs the rich to give to the poor herself.

3. Steals Bruce’s sports car.

4. Gives Bruce’s prints to a badguy, which bankrupts Bruce and makes him lose his parents’ company.

5. The loss of the company leads to the destruction of Gotham, with the usage of gadgets that the badguys stole from the Applied Sciences division (also, they got a nuclear bomb out of the deal).

6. She hands Batman to Bane, so that Bane can kill him. Instead, Batman’s just crippled and left for dead.

7. With Batman gone, Bane enacts Martial Law and destroys the entire city, so Batman can see the depth of his failure.

8. Batman gives her an escape route (with the clean-slate thing she had been striving for) from Gotham. She comes very, very close to taking it, but she returns at the absolute last possible second to save Batman from Bane, by shooting Bane. With a gun (which Batman expressly told her not to use when he first met her).

9. Batman decides to sacrifice himself to save the city of Gotham. She says, “Yeah, okay. Good luck, bro!”

10. He runs off with her and lives happily ever after.