It's drama, and Bruce likes it :)
Bruce represents high society and the rich, and Selina represents the street hustler and poor. The very people that Batman wants to protect. That's way at the end he gives his home to the children's group. She did steal the necklace, but when Bruce is at the dance with her, he tells her it looks better on her anyway, but that he still couldn't let her keep it because of sentimental value. So she took his car instead. It played out more like a lovers quarrel then a robbery. She also didn't help Bane steal from the stock market. When Batman asks her what Bane wanted with Bruce Wayne's finger prints she answered that she didn't know. When they are in the tunnels looking for Bane, so she didn't really betray him cause Batman was looking for a fight. When the gate closes, Batman turns to her and says "You don't have to do this" and she says "I had no choice, he would have killed me" (or something like that). She stays and watches the fight. It's clear she regrets what she did. When Batman returns from the bit, he doesn't really trust her, he bribes her using the software on a USB key. He needs her to take him somewhere, and after she helps him, then he trusts her with his Bat pod. She tells Batman that she's not staying, but after clearing the tunnel she returns to help Batman. The point isn't that Batman trusts her, but that he sets an example on what it means to be a hero. So she returns, kisses him and tells him she wishes things had been different. We then see them together at the end.