Okay, this is post-story-timeline entirely, because some things are just too spoilery.
1. Slughorn is a very old man, and this is the second Dark Lord’s rise he’s lived through, (and the second Dark Lord he unwittingly mentored) and he is retiring, Minerva, really that’s it, no more. I don’t care if you don’t have a replacement lined up. Mai, who for various reasons that are spoilery, at this point can’t live with her parents, takes the job as a temporary measure until they can find someone better suited. Somehow they never do. Then McGonagal retires and Flitwick refuses the job, and poor Neville gets stuck with the thing, and he makes Mai’s appointment permanent three years into her being hired, and drops the Slytherin Head of House job in her lap too. Mai gets burned hard by Sughorn’s favoritism as a student and deeply resents him for it at the time, so she’s scrupulously fair as a teacher. She is also about half the school’s favorite teacher, because some days she walks in and looks at her students and says “Today we are blowing things up.”
2. Ty Lee and Suki both become professional athletes, in Quidditch and Football respectively. The year Ty Lee retires, she plays Seeker for Ireland and they win the World Cup. Both sisters are heavily involved in sports and children’s charities on their respective sides of the Wizarding Muggle divide. For a long long time, if their partners aren’t around, they share a room, because both of them have horrible nightmares from the war, and they feel better if they know their twin is there.
3. Aang wanted to be a professional Quidditch player, but he gets sidetracked by his wife’s efforts to reform Wizarding government, and ends up running for the Wizengamot and winning. He’s a war hero, so it’s not surprising. His pet causes are undoing the purebood hierarchy, squib rights, greater governmental accountability, and actually working to eliminate the global magical antiquities theft problem.
4. Katara becomes a healer and advocate for indigenous magical communities. She travels between the UK, where her husband is in office, and Canada where her family is from. She regularly skirts the edge of the International Statute of secrecy as she fights to get medical care and clean water to First Nations communities in Canada.
5. At the same time Mai is hired as Potions Professor at Hogwarts, Toph and Zuko are trainee Aurors. After the war with his sister, Zuko and Toph are eminently qualified dark wizard catchers, but Zuko is burned out on it. He uses his status as Auror, to with his boss’s approval, set up Wizarding children’s services. This is something near and dear to Harry’s heart too, for obvious reasons. In the end, Zuko runs for the Wizengamot on a pro-getting-wizarding-social-services-for-fuck’s-sake-also-I’m-a-war-hero platform and wins. He and Aang are parliamentarian contrarian buddies. He and Mai turn the Nott ancestral home into an emergency foster care center, and pretty much always have magical kids underfoot. Because she’s a teacher at Hogwarts, Mai is teaching or will teach nearly all of them. A few of them are there because she reported their parents.