I'm going to try answering some of these questions.
1. I'm guessing just good old fashion detective work. He has an army of goons and plenty of resources. It's not impossible to figure out Batman's identity. Hugo Strange did it in Arkham City.
2. Assuming the Batcave is under Wayne manor I'm guessing Bane just stumbled upon it when he raided Wayne Manor.
3. This is harder to answer. The scene in the hotel with the Joker gave me the impression that Bane really didn't do it for the money. But then again according to his secret tape his goal is to find alternative to Venom to quell his addiction to it.