In the character introduction for the protagonist the is a shot from the ground, betraying the rubbery nature of the prop gun by the characteristic wobbling of the barrel. This makes for unintended comedy but is, of course, very sensible from a safety point of view.
As Joseph Black Moon is baptized, the rest of the congregation is singing the hymn "Pass Me Not, O Gentle Savior". The show takes place in 1865; however the lyrics to this hymn were not written until 1868, and the music until 1870.
In 1865 the completed capitol dome is shown but it was not finished until 1866.
While on the train, the man asks Cullen if he is a gunslinger. "Gunslinger" wasn't a term used in the time period. Gunmen or less common gunfighter would be the usual terms used at the time
Lily and Robert are shown at various times in fields of "wildflowers". One of those is a cultivated field of canola (uniform field of yellow flowers) which is certainly not consistent with 1865 Nebraska since canola was not even developed until the 1970s. At another point they are in a field of meadow flowers - most of which are introduced weeds including canada thistle, meadow brome grass and alfalfa, not native prairie.
When the forward survey team is attacked by Indians there are no guards at the camp, and no one seems to be armed. With all the money they were getting from the government, and all the money they stood to make, the railroad companies always hired armed guards to accompany their survey teams. In fact the Union Pacific Railroad Company hired guards from the Pinkerton National Detective Agency to protect their workers, especially when they were going into hostile areas like Indian territory.