Oh yeah, I've never watched the show and I'm enjoying this quest regardless of that fact but could someone please help to clarify what even is the relationship between Azula, Ty Lee and May? I just googled it but I'm more confused now.
Ty Lee and Mai are noble kids who a young Azula decided to "befriend" and train into minions. Ty Lee hung around because she was happy to have people who knew her and not her identical sisters, and Mai had a crush on Zuko (and both of them feared Azula, though that goes without saying). Shortly before the start of the show, they were separated; Ty Lee had ran away to join the circus (no, really), and Mai's father was awarded governorship of a conquered Earth Kingdom city (Omashu) so she was forced to move there. When Azula was tasked with hunting Zuko, Iroh, and the Avatar, she hunted down Ty Lee to intimidate her into joining the mission, and from there they went to Omashu to recruit Mai. Who immediately joined without even needing to be told what the mission was, because she was so incredibly bored.
Throughout most of the show, they more or less go along with everything Azula says (because it was either interesting/fun or out of fear/respect/friendship), but during the events of the Boiling Rock (specifically, shortly before the point where this Quest's timeloop starts), Mai betrays Azula and saves Zuko's life (who had at that point went full traitor and was helping certain high-profile prisoners escape). At the climax of the arguement:
"You miscalculated," Mai says. "I love Zuko more than I fear you."
Azula, who at that point was getting extremely stressed with Zuko "abandoning" her and Mai having just revealed that she "cared more about Zuko than her", finally snaps and tries to murder Mai. Ty Lee (who had been on the sidelines trying to talk them (Azula, mostly) down) intervenes and chi-blocks Azula before trying to escape with Mai. They fail, get captured, and Azula decided to just let them rot in prison instead of killing them. Their betrayal of her contributes to Azula's eventual mental-breakdown, and even after Mai and Ty Lee are freed their relationship with Azula never recovers. It is worth mentioning that they meet again after the main story ends in the comics (surprise, nobody is happy and it ends poorly), but nobody actually
likes the comics.
As for the other possible meaning of your question, no, they're not dating, though Azulee remains one of the most popular of Azula's potential ships.