At the end of season 3, Alice Morgan asks Luther to get rid of the coat. John throws his coat from a bridge in London to Thames. In season 4, the coat re-appears on John. Later in the episode, however, John pulls a drop cloth from a rack of clothing, which reveals several identical jackets.
When Roberta Wise presses the elevator button, the light goes on and then off again. This could be because it's a prop, or because the elevators were shut down for filming, or it could show that the killer had access to the elevator control panels.