When the Kid visits Jane, he ties his horse to the bar in front of the house. A short time after that, Ned arrives. He's searching for the Kid. He is so eager to get him that (later in the movie) he even shoots at him (when the Kid is fleeing from the court house). But when Jane tells Ned that the Kid is not there, he believes her without asking about the horse, which he must have seen when he arrived.
In the bar room, shortly after Whip McCord tells The Oklahoma Kid to give him back the Indian money, the Kid shoots Curley in the belly. A few moments later, Curley walks out as if he had never been shot.
Tulsa was founded by Native American tribes in, at it's latest, 1836...more than half a century before ANY of the Oklahoma Land Runs. Every element of the plot, including the statement that the city would be formed at the end of the same day of the start of the Land Run, indicates that the movie is actually about the birth of Oklahoma City, not Tulsa.
The goof item below may give away important plot points.
The Kid shoots Indian Joe a short time after Joe and McCord's (three) other henchmen have lynched his father, so there's no need to print a 'wanted' circular for Indian Joe. And when the circulars for the lynchers are hung up, there is indeed no circular for Indian Joe. However, later in the movie, when Judge Hardwick and Alec Martin talk about the Kid, the Judge has four circulars, including one for Indian Joe.