Did the Air Nomads have a military?
The Air Nomads had no formal military , but sexual segregation was a major part of their society. Men and women inhabited entirely different Air Temples in the days of the Air Nation, and integration and cohabitation among the sexes appears to be extremely limited. When the nation was reborn in The Legend of Korra, the archaic practice was disposed with entirely, and barely even addressed.
Is there a woman in the Fire Nation?
It's possible that there are other women among Earth Kingdom's armies we simply do not see, but the fact they are unseen speaks to the Fire Nation's progressive ness when it comes to their ability to integrate. Princess Azula, Ty Lee, and Mai make up an elite force at the head of the military, and at every stage below, there are many women serving. The prison guard Ming even receives a name and dialogue with Iroh, fleshing out her personhood and the presence of women in the story.
Was the Avatar the only one to restore balance to the world?
The Avatar may have been the only one to restore balance to the world, but as where gender equality was concerned, it was the Fire Nation most successfully striving for equality.
