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.
Other mistake: While playing as the Joker, despite his claims that he'll kill all enemies he comes across, if you zoom in on their corpses, you'll notice that they're still breathing, regardless of how you take them out (take-down, shot, or blown up). It's easier to spot this mistake in the invisible predator challenge maps.
Audio problem: When Batman types on the computer in Dr. Young's office, the clacking sounds are woefully out of synch with his hand movements, not to mention his fingers never actually touch the keyboard.
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.