She could have been Rachel Dawes's sister and out for vengeance and it would have meant the same. Who she was didn't matter. That she was about to blow up a nuke and destroy Gotham was what was important.

As for why Wayne trusted her? It was because that's what the movie needed. It doesn't make sense. It's like asking why, if Bain's goal was just to upload a program that could fake trades into the stock exchange, would be shoot his way in and let everyone see that he was uploading something. It's because the movie needed some action scenes and a chase.