When Frank Griffin ties his arm in back of the saddle the fingers are facing to the right side of the horse. When he rides into the church, with no indication he and his men had stopped, or camped, no reason for the arm to have been removed, the fingers are now facing to the left of the horse.
The money in the payroll is all currency. Miners in the west wouldn't have accepted currency, or "paper money" in those days. Gold and silver coins were needed to pay workers in those years after the Civil War during which both US "greenbacks" and Confederate paper money had been devalued by being over issued. Working class people didn't trust currency in those days which is why there were more US Mints west of the Mississippi than in the east.
As a character is speaking of a narrow gauge railroad, one of the overhead shots is of a standard gauge, not narrow gauge.
When the Marshall is being told how the men of the town all died, the story was that "there was too much coal dust" in the mine and that an explosion was caused by a spark. The mine isn't a coal mine, so there wouldn't be any coal dust to cause an explosion.
Sheriff McNue asks Roy Goode where his 'black handled .38' is, in this period guns of that calibre were known as .36 or Navy calibre. The bullets are the same diameter but firearms nomenclature has changed in the intervening time.
The mini-series takes place in 1885, and the Marshal orders Bonded Whiskey. The Bottled in Bond Act was passed in 1897. Bonded Whiskey wouldn't have hit the market until the early 1900's.
At about time line 27:21 the woman smoking looks to be smoking a modern cigarette.
When Roy Goode lay down the horses. There is a rope visible to help lay down the 2nd horse. When the horse is on the ground the rope is removed from the right side of the screen.
While tracking Roy Goode, Frank Griffin is told by the tracker that Goode possibly went down the Purgatoire River to New Mexico. The Purgatoire River's headwaters are in the Colorado Rockies, and it flows northeast into the Arkansas River. It never flows to or from New Mexico.