Continuity mistake: When you find Gordon in the mall he's wearing his jacket. When you rescue him just after your final fight with The Arkham Knight he's not. Then he's got it back later at the asylum.
Batman: I am vengeance. I am the night. I am Batman.
Trivia: When Batman enters the clock tower, there's a flashback scene of Barbara being shot by the Joker. Before this occurs, go to the back wall and zoom in on the movie poster, "Dark Interlude." One of the names on the poster is Matthew Hagen. In "Batman: The Animated Series", Matthew Hagen was an actor who was transformed into the shape shifting villain Clayface.
Answer: As a child, Jervis Tetch never had any friends. Even in his adult years, he was all alone. When Jervis asked Batman to be his Alice, he was essentially asking Batman to be his friend.