In the movie, during the European tour, Annie shoots a cigarette out of the mouth of Crown Prince Wilhelm of Germany (later to become Germany's Kaiser). There was such an incident, but Annie didn't shoot the cigarette out of Wilhelm's mouth due to the danger but shot it out of his hand instead. During WWI Annie, reminisced that if she could do it over she'd let him put it in his mouth and then miss.
The shooting contest between Annie Oakley and Frank E. Butler ("Toby Walker" in the film) was actually won by Annie Oakley, not by Butler ("Walker") as depicted in the film.
Although in 1935 gun safety topics were a much tinier subset of what they are today in 2023, the Toby Walker character at 34m10s into the film bundles up five (5) of his shotguns or rifles into both arms, across his belly, and carries them with the business ends pointed at / to / level with everyone around. When one walks around with firearms of any sort, they should be treated as loaded, and generally only pointed up or straight down at the ground, never at others; also, because the character is transporting guns to his new sleeping quarters, a firearm owner today, by the best practices of gun safety, would employ the use of firearms cases.
In the final scene when Annie is shooting the targets you hear nine rifle shots, but you see sixteen targets destroyed.
Annie returned to New York in 1892, and the film has the band playing "The Red Man", from "Dwellers in the Western World", which Sousa didn't write until 1910