One of the biggest problems with looking for historical inaccuracies in Mulan is that the movie intentionally unmoors itself from time and place. Its nominally set in the Han era, but uses a post Han era enemy (and one that never directly fought China), while using a lot of much later styles (Song Dynasty), architecture (Ming or Qing), and weapons (Song again). It isn't set in a particular time. It's set in "mythic ancient China" and seems to include elements from a millennia and a half of history.

It's a little like having a movie where Julius Caesar and Joan of Arc load their troops onto Portuguese caravels to defend Elizabethan England from Viking incursions. When would such a movie be set? If it didn't give a date, how could you date it? How would one even start to discuss the historical inaccuracies?

It's also worth remembering that the various versions of Hua Mulan are set in differing time periods as well. In the Ballad of Mulan the setting is Northern Wei. The much more famous (and much later) Sui Tang Romance sets her in the Tang era. So being historically inaccurate is actually historically accurate from a certain point of view.

what would actually happen if 1) a girl actually did that

That's an interesting question. I'm not fully enough versed in Chinese history to give a direct answer, but I can recommend a couple of books

• Women in Chinese Society by Margery Wolf - This mostly focuses on the later Chinese periods 18th, 19th, and 20th centuries. But it references earlier periods a fair amount.
• Women in Early Imperial China by Bret Hinsch - This covers women of the Qin and Han periods.

Unfortunately I don't know of a book that focuses on women in the middle period in between.

the Huns

My guess is you're going to be happier if you just translate Hun to barbarian and consider them a pastiche of steppe tribal culture. If you really wanted to compare "the Huns" of Mulan to the historical Huns, you'll probably want to read something like:

• The World of the Huns: Studies in Their History and Culture by Otto Maenchen-Helfen - Its a bit long in the tooth, but it does a good job of discussing culture rather than just a list of wars.
• The Huns by Hyun Jin Kim - This is much newer, and I haven't read it yet. But it comes highly recommended and covers groups like the Xiongnu which would be the "Huns" encountered by China.