Corrected entry: Throughout the film, Hatsumomo appears many times with her hair down. Although the director states in the DVD commentary that he wanted to portray Hatsumomo as a 'free spirit' and hence the hairstyle, this will never have happened in Kyoto of those times. Most geisha back then would style their hair with hot wax, and would only go to the hairdresser once a week, and wear the hairstyle day and night. Therefore, no geisha was seen walking around with her hair down, even during the day.
Corrected entry: During the first parts of the movie while, Chiyo is still a young girl, she is brought to live at the okiya. Notice that her friend Pumpkin is tacking a "chan" on the end of her name (i.e. Chiyo-chan) This is a Japanese honorific that girls use for each other. When you first meet someone, even after you have known the person for a while, you still refer to them with the honorific. You only drop the honorific, and refer to the person by just their name ONLY after you have known them for a while and have asked permission. Pumpkin uses the honorific, but Chiyo does not. Failing to use this honorific when you should is very rude in Japanese culture. It can't be that she does not know this because at the very beginning of the movie, she calls out to Mr. Tanaka, saying "Tanaka-san" (the Japanese honorific equivalent to saying Mr. Tanaka).