In at least one episode, when Walt finds a character in possession of a firearm, they explain to the sheriff that "it's registered". Wyoming, like many states, does not require registration of either firearms or firearms owners.
Walt Longmire is shown to be subservient to the mayor of the town. As a sheriff, he would answer to county commissioners, not a town mayor.
Several major characters are shot with rifle rounds throughout the series, yet they not only recover, their wounds are not shown to cause them any long term injuries.
Several times throughout the series, buildings are shown with a southwest US architecture in the Pueblo Indian style. This is a trait of towns in New Mexico, where the series is filmed, not Wyoming.
The setting of the series, Absaroka County, Wyoming, while fictional is also well over 100 miles from the real life Northern Cheyenne Reservation, which is located in the neighboring state of Montana.
Throughout the series, Walt never has more than three deputies, giving his department a total of four sworn officers. In later seasons, he only had two deputies. Even in a small rural county it would take more staffing than that to provide 24-hour coverage. This error is compounded when every deputy on duty accompanies Walt on the case of the week, meaning there is no coverage for the remainder of the county.
Several characters, on several occasions, refer to the "high altitude" of the area. They actually mean "high elevation". Elevation refers to the height of a point or place on land, relative to Sea Level. Altitude is a measure of the height or distance of an object in the air from the ground.