In the opening scene, when John Dutton has to shoot the horse, the horse's dead body is missing as John is walking away a few seconds later. The horse's dead body would have clearly dropped where it was shot.
The "wild" horse is wearing horse shoes.
When John Dutton and Lee Dutton are helping the cow birth her calf, Lee pulls the calf out, the men stand, and both cow and calf get up and run. This is not how that works. The cow and calf would just lay there, recovering. A calf cannot stand that soon after birth, plus the placenta would still be attached.
When the friend asks what medal he is pointing at, Kacye's BiL answers that it is a Silver Star. In fact, the medal being pointed at is the Bronze Star. There is a Silver Star in the shadow box but it is three medals to the right of the one being pointed at. Also, generally medals are arranged in a shadow box in order of precedence, starting in the upper-left (as observed) corner. Both the Navy Cross and Silver Star in the shadow box have a higher precedence than the Bronze Star.
In the pilot episode (1.1) when everyone is looking at Kayce's medal rack from his Navy service, his friend is pointing and asking how get got a "Silver Star Medal" when he is actually pointing at a Bronze Star Medal (the silver star was several medals away to the right.
Kayce made it seem like the Silver Star was a higher honor than Navy Cross. In fact, the criteria for the Navy Cross more rigorous than that of the SIlver Star.
Casey lifts fish cooked in aluminum foil away from hot coals. As he lifts it and moves it towards his son, he tells his son to be careful, the top is hot. He then moves the hot fish in the foil over to his son's leg.
At 1:27:17 there is a close up shot of Kevin Costner wearing sunglasses, with a cameraman clearly reflected in them.