Dr. Ogden explains the victim's different colored eyes as heterochromia, "usually a genetic phenomenon". The term "genetics" was first used in 1905, and the story of this episode takes place ten years earlier.
The inspector uses the expression "mad as a box of frogs." This expression was first used in the second half of a the twentieth century, and the events depicted here take place in 1895.
When the dummy is insulting Arthur Conan Doyle, Murdoch instructs the police officer to unlock the cell door. He (Murdoch) then strides in, seizes the dummy and walks out. The police officer slams the door shut but does not use the key to lock it again. Since all police cells have deadbolt locks, this would mean that the cell is left unlocked.
When Murdoch asks the doctor for the time of Grimesby's death she replies "between 4 and 5 a.m. this morning." She should have said either "a.m." or "this morning", not both.