Right before the shoot-out Rip throws his duster back and puts his left hand behind his back (presumably to keep it out of the way of his gun), but the shots from behind Rip show his left hand in front of his body and the duster hanging free, obstructing his draw.
In 1871, there were no plastics. If they were really trying to fit in with the locals when first arriving, Dr. Stein should have traded his plastic frame glasses for wire frames.
Ray says that the six shooters in the Old West at this time were actually 5 shooters because they had hammer blocks. This is incorrect. The revolvers at this time did not have hammer blocks to prevent the hammer from hitting and firing a bullet accidentally if a gun was dropped. The guns did hold 6 bullets but the chamber under the hammer was usually left empty to avoid accidental discharge so most guns were loaded only with 5 bullets..
When speaking to Ray, Martin refers to him as "Brandon." This is the actor Brandon Routh's name, not that of the character Ray Palmer.
The Colt Single Action Army carried by Sara wasn't produced until 1873.
Rip tells Stillwater that he will be shooting for "Sheriff Palmer" and Stillwater understands despite knowing Ray Palmer as "John Wayne".
The town razed in Rip Hunter's previous visit to the past was said to be in Oklahoma. At that time, neither the state or name Oklahoma existed. It would have been Indian Territory.
At 06:55 -Martin places a bet before he's picked up all his cards.