These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
T'Pring nasty, manipulative villainess or justifiably angry Woman Scorned? Spock has been ignoring her and their marriage for roughly twenty standard years, basically humiliating her in front the entire planet. In the meantime she's found a man who loves her and treats her right and doesn't want to risk his life.
When Spock tells T'Pau that he will "do neither [live long or prosper]", did he mean he was suicidal or did he expect to suffer Death by Despair over Kirk's murder?
Angst? What Angst?: Kirk and Spock are both awfully calm after their reunion considering that Spock almost killed Kirk.
Ear Worm: The infamous fight scene music, which became so well known that Eddie Murphy referenced it in one of his stand-up routines.
Fanfic Fuel: Fanfics have been penned on how young Spock and T'Pring felt about the Arranged Marriage and how Kirk and Spock coped with the aftermath of the kal-i-fee.
Kirk gets ridiculed for jumping from "biology" to "reproduction", but Spock is obviously embarrassed and what part of biology is half so embarrassing to talk about as reproduction?
Also, T'Pau says that T'Pring will become "property" of the victor. Presumably, Spock could have treated her much worse than he did and been within his rights... just like T'Pring was in pitting Spock against Kirk. She's lucky he didn't decide to go Beware the Nice Ones on her.
In Spock's final exchange with T'Pau, he responds to the traditional "Live Long and Prosper" with "I shall do neither." Did Spock mean that the civilian courts would execute him, was he contemplating suicide, or did he mean he would die of grief and guilt?
Also, T'Pring explains that she could do whatever she liked because Spock would be "gone". Did she mean "away on a mission" or "dead"?
Also, the end. Yes, Spock won't kill himself or die of grief now that he knows Kirk is alive, but he'll have the guilt of having attempted to kill his best friend for weeks, maybe the rest of his life.
Also, the fact that the most logical, (theoretically) sensible and smart people in the Federation only allow people to back out of arranged marriages via fights to the death, and that the woman in such cases becomes the property of the victor - What the Hell, Hero? barely covers this!
The episode tries to wring tension from two main characters possible deaths (and the supposed death of one). Given that the show is from the era of strict Status Quo Is God, the ending is never in doubt.
Also, Kirk sacrifices his career to save Spock. Yeah, that'll stick.
Magnificent Bastard: T'Pring. Spock is quite right in saying her logic is flawless. No matter which way the combat went, she would end up with everything she wanted. And the fact that she ends up as a Karma Houdini (so far as the episode goes) makes her victory that much sweeter.
Throw It In: The Vulcan salute makes its first appearance in this episode. Leonard Nimoy came up with it on set. Incidentally, Celia Lovsky, the actress playing T'Pau, couldn't do the salute and had to arrange her fingers with her other hand off-screen.