Other mistake: The doctor is supposedly being led away in handcuffs, but isn't handcuffed when an explosion goes off behind them and they are thrown to the ground. (00:37:30)
Continuity mistake: When Agent Keen attacks Craig in his hotel room he pushes her to a rectangular shaped mirror on the wall but when Red enters the hotel room they zoom in to the same mirror, which is now square.
Answer: The man Red killed, Sam Scott, was Lizzie's adoptive father, not her biological one. However, at that time Lizzie (and thus the audience), was led to believe her real father abandoned her and her mother, so expect more secrets to be revealed how her real father actually died (or if he's even actually dead).
Bishop73