Kate's hand positions change between shots when she's pulling Castle with the rope.
The stagecoach that Castle and Beckett arrive at the dude ranch in says 'Spearfish - Deadwood Stage'. The stagecoach must have been borrowed from South Dakota. Since the dude ranch is in Arizona, it should say something like 'Tombstone - Agua Caliente Stage'.
As this show's time is post-1968, NY residents Rick and Kate would not be allowed to buy real handguns in Arizona.
Det. Ryan says "Whitney was looking up obscure documents and maps from the late 18th, early 19th century." which would put it in the late 1700s-early 1800s. That would be about 100 years too early for the Peacock Brothers, or the damming of the stream mentioned as "in 1920".