When the crowd of civilian spectators is stampeding during the battle of Bull Run, some of the female extras can be seen wearing knee socks, and not the period petticoats and long drawers that women wore in 1861.
In 1861, the United States Capitol in Washington is shown without any dome, but work on the new (and current) dome was well underway at this time, and scaffolding would have been visible, along with a construction crane.
The map on the wall in Lincoln's war room have Union troops denoted as American flags with 41 stars. At the time of the First Battle of Bull Run at Manassas, 21 July 1861, there were only 34 stars on the Union flag. At no time in its evolution have there ever been 41 stars on the US flag.
Just before the war begins Billy Hazard tells Brett he is joining the U.S. Sharpshooters and subsequently joins the unit and fights in the First Battle of Bull Run, which occurs July 21st of 1861. In fact the Sharpshooters were not authorized until September of 1861 and did fully muster until November. Billy could not have applied to the regiment until at least two months later.
After the first Battle of Mannassas, when George Hazard is lamenting the outcome with Lincoln, he states that 28 cannon were lost, 17 of them rifled. Rifled cannon where not used until April 1862 with the bombardment and fall of Fort Pulaski near the mouth of the Savannah River.