When reading from the newspaper, Charles quotes "Governor David P. Applewood", however there was never any Governor of Minnesota of that name. At the time (1876), the Governor was John Pillsbury.
The town talked about celebrating July 4th on Sunday after church, however, July 4, 1876, was on a Tuesday.
Walnut Grove is getting ready to celebrate the Centennial of the United States which means this would be in July. However, they have the kids in school which they wouldn't have been. Children didn't go to school in the summer.
Mr. Pyatakov said he had a homestead waiting for him in South Dakota. In 1876, this would have still been Dakota Territory. South Dakota wasn't admitted into the union until 1889.
At the end of the episode as fireworks are being shown, "America The Beautiful" is heard. However, since the episode takes place in 1876, the song would not have existed. The tune was written in 1883, with the lyrics, taken from a poem by Katharine Lee Bates, added in 1895.
When Mary is finishing the flag near the creek, prior to July 4th, 1876, she tells Laura that she can't decide whether or not to sew on a 38th star, for Colorado, which joined the Union on August 1st, 1876, almost one month after the Centennial. The flag at the end of the program has 38 stars on it.
Yuli states that he and his family have only been in the USA for 6 months, and that none of them knew any English until his son Viktor started school. Yet Viktor speaks perfect English with a perfect American accent, even when alone with his parents.