"Look at the life rabbits lead," Bugs says. "Dogs, hunters, hasenpfeffer." Hasenpfeffer is a traditional German stew made from marinated rabbit or hare.
Yes. He was famous for jumping off the Brooklyn Bridge in 1886 and surviving. In Bowery Bugs, we see a plaque on the bridge that misspells his name as "Steve Brody." See: the Wikipedia page for Steve Brodie.
It's a reference to the fabled scam about "selling" the Brooklyn Bridge to any gullible tourist. In this case, the old man bought the bridge after being impressed by Bugs's story of why Steve Brody jumped off it.