Hank Rusty Truman: Are they into public displays of affection?
Of course! She even calls him "Zuzu" in front of literally everybody. Yes, it is affectionate. Always affectionate. Including when she had just shot lightning at him. I will take no criticism. Fuck you, fight me. "Dum-dum" is also a pet-name, and Zuko will call her "Lala" and "Zula" in front of others - half the time he does it to get on her nerves, and the other half he's just so caught up in being all sweet to her that he forgets there are people around. And obviously, calling each other by formal titles like can be sweet and flirty, and don't even get me started on how they make the words "brother" and "sister" sound dirty.
Some of their displays of affection are subtle, and could be mistaken as purely platonic/familial. A hand on the shoulder, head-pats, hugs, always sitting right next to each other, praising and standing up for one another, bickering and teasing (that IS one of the ways they show affection to each other)... Zuko also decides to play the role of a gentleman in public (only in public 😈😈😈), pulling up a chair and holding the door for her, and he also has her be the first woman he dances with at every ball, festival or party.
Other things they did, however, made people uncomfortable long before they had undeniable proof that, yes, these two are definitively lovers. At every ball, the only other women Zuko would ever dance with were Katara, Suki, and Toph, and these dances always looked far, far, far more innocent than the ones with his sister - who would simply not dance with any man. They'd constantly have these private conversations in front of everyone, whispering to each other and chuckling, and then they'd either leave the room holding hands/with arms entwined, or one of them would make an excuse and leave, with the other, who was obviously staring at them as if under a spell, making up an excuse to leave two minutes later. Their hugs last waaaaay too long, they're far too protective and possessive of each other, and praise each other waaaay too much.
And, of course, we have the things they do when everyone already knows about them. Azula will sit on his lap at every opportunity, and will even make some very inappropriate comments about how "loving and selfless, but rough and demanding" her big brother is. They'll kiss and full on make out in front of anyone that happens to be in the room whenever they want to get a taste of each other... which is all the time. It is also not a good idea to watch them practice their bending and spar with each other unless you want to (or is at least willing to tolerate) a hot, shameless, highly incestuous fight that is just guaranteed to turn into sex (because fighting IS one of their ways of demonstrating love and lust)