Paladin ties his horse to the back of a wagon. He never takes the saddle off the horse, when in reality, he would have.
After Paladin ties his horse to the back of the wagon, his horse disappears in distance shots of the wagon, and reappears behind the wagon in other scenes.
At the conclusion of the scene where Paladin helps the stranded family with their broken wagon wheel, as the wagons leave there are very obvious wide, modern-day tire ruts on the dirt road/trail.
Paladin is driving a wagon with a spare wheel tied to the side. The wheel is larger than any of those used by the wagon. Paladin and his passenger happen across a family stranded with a broken wheel on their wagon. That broken wagon does require the larger wheel. The wheel is installed on the wagon, the family is rescued, and Paladin goes on his way. Notwithstanding the improbability of carrying the unneeded wheel, this incident has nothing at all to do with the main storyline of the episode, other than to point out that Paladin is a man of character who is unwilling to leave a family stranded, and that Paladin uses the incident to establish who is calling the shots on this expedition - himself, not his client.
Paladin would never use his valuable handgun as a hammer as when he does when he finds the Gatling gun and ammunition. He uses the butt of his gun to damage one of the clips for the gun. His gun is supposedly perfectly crafted and balanced.