Adding to my question from earlier, I forgot to mention that my character is male. Also, are there any ballet styles that involve acrobatics such as flips or rolls?
The American school of classical ballet is renowned for more lively and powerful performances, but your description makes me think of Neoclassical and Contemporary ballet, which happen to include a lot of spins and flips. You know what else includes a lot of spins, flips and jumps? Folklore dance! (also called character dance)
Fans are traditionally used in many folklore pieces, as well as in classical variations like in Don Quixote, The Nutcracker etc. but always by women, so I don’t think it can apply to your character and at the moment I can’t think of any piece where men use fans.
If your character can manipulate air, he could use his power to land incredibly high and complex jumps or turn faster. I think it’s up to you and your character to figure out which style of dance best works with his abilities, personality, and weapon, it also depends on the connection that you decide to create between dance and fighting. Whatever the style, anyway, dance will definitely improve your character’s coordination, physical resistance, and agility.
Hope this helps!