Other mistake: In the side mission "Identity Thief" Hush leaves Arkham City after revealing he now looks like Bruce Wayne. Problem is how he's able to, especially with Penguin's reward for bringing Bruce Wayne to him.
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Suggested correction: Hush is a resourceful man, and could figure out a way to disguise himself or get to his escape route with minimal contact with anyone else. Plus, the Penguin's gang was decimated and Arkham City thrown into chaos during the events of the story, so most of the inmates probably had bigger problems than chasing a bounty that was likely rendered moot when Batman imprisoned Penguin midway through the story.
