When Elder Wiggs breaks up the fight between Sandy and Jackson, a dog joins in and tears one of the legs of his pants. This was not in the script. It just happened on the set, and John Ford had liked it so much he kept it in the film.
Harry Carey Jr. rode his own horse called "Mormon" and Ben Johnson rode a famous movie horse called "Steel" that was owned by his father-in-law "Fat" Jones, who ran the most well known horse renting stable in Hollywood. In the galloping scenes Johnson rode Steel's stunt double Bingo and was quoted as saying he was just a passenger as "Bingo" thundered down the hills. According to Carey, Steel and Mormon became very attached and ruined quite a number of scenes by calling out to each other.
In the scene where Sandy and the Mormon fight, the fight is broken up by Ward Bond's character. Bond has ripped pants as he separates the fighters and you can hear a dog barking in the background. This happened because director John Ford wanted to use two dogs, that had been ruining every scene in the film by fighting, in the background as the men fought, hoping the dogs would start fighting as a contrast to the men fighting. Instead of fighting, however, one of the dogs ran away and the other attacked Bond and ripped his pants. Ford could barely contain his laughter but kept filming. Afterward, he became quite concerned and said they needed to find the dog in case it had bitten Bond, not just ripped his pants. Ford was worried the dog might have needed a tetanus shot.
In the scene where Travis gets bucked off his horse after Denver throws water on it, Ben Johnson did his own stunts. They used a genuine rodeo bucking horse and John Ford promised Johnson if he rode the horse he would not have to do any dialogue for the day, which apparently pleased Johnson. He lasted four bucks and came off so hard he was almost knocked out. Unfortunately, the shot was ruined by one of the wranglers running out to him and asking if he was all right as he lay on the ground. Johnson had to get up and ride the horse again. This time he lasted ten bucks before he bailed off and Ford got his shot.
According to Harry Carey Jr., Joanne Dru's husband, John Ireland, stayed in town during the shooting and avoided the set, but did organize the company into a performance of "The Shooting of Dan McGrew" in the evening.