Étude for Firebenders - An étude is a classical music form in which the piece is generally short and deliberately difficult, often designed as an instructional exercise.

wartime Fire Nation colony - It is not clear, canonically, which Fire Nation colony became Republic City, though one rather tends to suspect it was Yu Dao.

use my body as a weapon - It has not escaped Anne that, in her new life, she is also arguably being made into a weapon, but she doesn't consider it ironic; it's of her own choosing, and - most importantly from her perspective - she wields herself.

depict her as a monster - One of those artists was Sokka, who was known (and occasionally suspected of preposterous Fire Nation sympathies) in his time for always faithfully depicting Azula as she had actually looked, even in images that showed her doing her worst. Many of the other contemporary artists on the anti-Fire Nation side at least slipped in some fangs and/or horns, though even they usually couldn't bring themselves to make her ugly, just monstrous.

Zuko's detractors - Including Azana's mother, Azera, who has always maintained that Zuko was a usurper and, in general, a failure as Fire Lord.

a bit in love with him - In the light of the above, this may have begun as an act of contrariness.

Kaiten - This character's name comes from one of Japan's desperation weapons of World War II, the Kaiten manned torpedo. As the name suggests, this was a naval torpedo designed to use a man as its expendable guidance system - quite literally a suicide weapon. (The name kaiten transliterates as something like "return to the sky" but was usually translated more along the lines of "ascent to heaven".)

Princess - How Kaiten came to know Karana's pet name for Azana is not certain; Azana has tried to find out before, but to no avail. Regardless, he is certainly aware that it is an intimacy to which he is in no way entitled. He's being deliberately offensive whenever he uses it.

your chunky little pal - Some guys just don't go for that buff Water Tribe look. (Those guys have pretty poor taste, you ask me.)

"Hah." - This is shorthand for, "On the list of distasteful things that have ridiculously been put to me as my patriotic duty, that isn't even number one. This amuses me in a dark and cynical sort of way."

the old man and the Avatar - Also a play they do every year at Republic City Primary School No. 4, though the title of the play refers to a different old man.

time for us to get acquainted - In what may be the only redeeming feature Kaiten ever had, he doesn't mean it like that.

Beifong - Common slang for metalbender cops (less insulting than "tinhead" and not as dated by this time), from the two members of that illustrious family who served as chief of police in Republic City's early days.

five-oh - Most Standard-speakers in Dìqiú don't know why "five-oh" is slang for the police, just that it is.

one more ashmaker - Like many of the doctors and nurses at Yugoda Memorial, Dr. Kuva is a waterbender. As such, he is not really inclined to give firebenders the benefit of the doubt, particularly members of the Agni Kai. I hasten to note that he hasn't sandbagged on his standard of care; that would be a grievous breach of professional ethics. He just expends no energy whatever on bedside manner when dealing with them. His oath requires him to treat them. It does not require him to give a damn about them in any other context.