Cpl. Agarn is from Passaic, NJ, which he pronounces "Pah-say-ik". The correct pronunciation--as anyone from Passaic knows--is "Pih-sayk".
Trooper Duffy claims to have been a Texas Independence fighter wounded at the Alamo. No white adult males from that army survived the battle. However, he is probably telling a tall tale.
James Hampton's character, Bugler Hannibal Dobbs, is a Texan serving in the United States Army immediately after the Civil War. Although white males from Reconstruction (formerly Confederate) states were under a blanket suspicion of treason at the time, many non-conformists proved loyalty by fighting for the Union during the War. It's possible that Dobbs was one of these.
The breech-loading revolvers and rifles seen in the show had not been invented as of the late 1860's.
The medal pinned on Parmenter's chest in the title sequence -- supposedly the Medal of Honor -- is in fact a modern-day Silver Star, first awarded in 1942.
Set in 1865-7, the series contains many references to the Battle of Little Big Horn (1876).
The show takes place from 1865 to 1867, yet the fort flag has 37 stars on it. But the 37th state (Nebraska) wasn't admitted to the Union until Friday, March 1, 1867, so the number is incorrect for most of the series.