In many of the stories when it shows a pigeon delivering a note to the train, this couldn't possibly happen. Pigeons only fly back to their home base when taken to some far off area. Pigeons would not know where to fly too from a home base. Especially to a train that could be anywhere.
Jim and Artie's train was missing at least three railcars in most shots. The cars that were not shown would be where they kept their horses, where the train crew was boarded and where the two of them slept.
The train car has numerous hidden compartments which slide out of the wall or down from the ceiling. In actuality these compartments are much too large to fit inside the walls of the car and would extend out past the sides and roof.
In many episodes there are sparks coming from under the train as it moves, usually towards the end of the episode, revealing the train to be an electric miniature.
Despite being in many stories that took them miles away from their train, Jim and Artie rarely carried enough supplies to camp out or survive in case something happened to their horses.
In the Old West this could have meant their deaths and experienced former soldiers would know this.
The United States Secret Service is frequently employed as bodyguards for President Grant. But this was not the case in the 19th century, when the Service was a Treasury operation used to catch counterfeiters. The duty of being presidential bodyguards was assigned to the Secret Service in 1901 after President William McKinley's murder. McKinley's immediate successor Theodore Roosevelt was the first chief executive to benefit from this change.
Most of the male characters were clean shaven. That wasn't the fashion style of that era.
Throughout the series various characters mispronounce "cavalry" as "calvary", confusing a pair of near-homonyms with Latin origins. Cavalry means soldiers on horseback. Calvary means an object shaped like a skull. The most (in)famous "place of the skull" (Calvariae Locus in Latin) was a Jerusalem landmark where Jesus Christ and various convicts were crucified during Roman Empire times.