Jason Todd had been the Robin the Joker had killed. Then, Jason was resurrected by Ra's al Ghul which caused Jason to go insane and become the Red Hood.
I think by the time of Man of Steel, Batman had hung up the cape and cowl and the Joker was already imprisoned. With the Suicide Squad movie, Waller uses the Joker to lure out Batman as the Joker was the only one who had ever seen him. It is implied in Batman v Superman that Lex Luthor is manipulating the heroes into fighting each other to show how bad they are and to fuel the fear and hate for the superpowers. It seems there are too many wandering subplots within the movie which detracts from the main story. How the three heroes band together to fight a big evil.