Question: Why didn't Batman just contact Gordon and tell him what was really going on in Arkham City? They could use that against Professor Strange and stop Protocol 10 before it even began.
Answer: Because Batman didn't know until it was too late, and there was interference as he spoke to Alfred, if you remember hearing that. So contacting Gordon would've been even harder.
Question: How did Hugo Strange find out that Bruce is really Batman?
Answer: Listen to the fourth Riddler interview tape. He did a psychological profile of Batman, then picked among candidates as to who could be Batman. He's also smart enough to realize only a rich man could afford all of Batman's gear and vehicles.
Answer: Batman later in the game discovers that Hugo Strange has been supplying everybody with weapons, so he would have given Two-Face the vat of acid.