Other mistake: When you enter the caves just after Killer Croc tells you he's going rip you apart, if you try to use the grapple, the grapple will not stick - Batman will say that the place is falling apart, and you need to climb manually. You can't grapple any of the edges, yet, if you fall into a pit and press RB to "Escape" like most other parts of the game, he'll grapple up to the most recent edge, despite his previous claim.
Trivia: After Dr. Young is killed in the warden's office, go back down the stairs but do not go through the archway. Instead, after going down the steps, turn right and you will come to a blank wall. Spray the explosive gel on the wall three times and then detonate it. The wall will break apart and you will find a secret room. Go inside and you will find blueprints for the Arkham City Prison Complex.
Question: In the comic books, animated series, etc, was there a passage connecting the facility to the bat cave like there was in the game? I thought that the bat cave was further away, like under Wayne Manor.
Answer: It's not the actual Batcave, it's a secondary location Batman set up in case of emergency. Oracle actually makes a comment along those lines the first time you access it.
Answer: To add to the answer, Batman tells Oracle that he found the cave after saving an inmate from committing suicide and had secretly been moving supplies to the cave after discovering its location.