Contemporary newspaper reports after Lincoln's Gettysburg Address describe his speaking voice as high-pitched and sometimes "scratchy", not the booming voice of an accomplished orator one would expect. The Lincoln portrayed in this series by Hamish Linklater is accurate as to the President's speaking voice.
Lincoln's nickname for Stanton is Mars. Stanton is the Secretary of War. Mars was the Roman God of War.
Tobias Menzies' casting is age-appropriate: The actor is 50, where Edwin Stanton was 51 at the time the story takes place. The two men look nothing alike, however - the real Stanton sported a long beard and wore spectacles.
There is no documented use of "Hail Mary" as a term for a long shot until after 1900, well after the trial depicted in the series.
John Wilkes Booth's favorite part he played on stage was Brutus in Julius Caesar because he "assassinated a tyrant". Tobias Menzies played Brutus in HBO's Roma (2005).