Ty Lee doesn't give a damn and just goes along with what Azula wants and Azula just sees Mai as a form of control over Zuko. Mai was always the least free spirited of the bunch and heavily tied to the nobility of the fire nation making her the ideal pin to hold Zuko down. Contrast Mai with Ty Lee who is free-spirited enough to run off to join the circus dispute it being unbecoming of a noble, if she tried to do the same thing using Ty Lee the two would run away together.
As for the Zucest angle, there's a bunch of things like the bedroom scene for example but the main thing is that Azula looks at Zuko as hers even if he is with Mai, it's clear that to her he is as much property as he is family since she treats his disobedience to her like you would treat a misbehaving puppy. He's the the only man alive that not only loves her but is also willing to stand up to her and given what a sociopath she is it's not far fetched that she would use ANYTHING to control him.