During the battle of Sekigahara, it was either partly cloudy or cloudy. According to Japanese records, on the day of the battle it was foggy and there was also some rain happening as well.
The narrator made numerous errors at the end of the show. First he stated that Ieyasu earned the title of the Shogun after he defeated Hideyori, the Tokugawa Shogunate was established in 1603, and the Seige of Osaka Castle was in 1614, 11 years after the Shogunate was established. Second he stated that the Shogunate would rule unchallenged for the next 250 years, again false. In 1638, Japanese Christians rose up in revolt against the Shogunate in Shimabara, sparking the Shumabara Rebellion.
The castle that is mostly seen is not Fushimi Castle, it is Himeiji Castle.