Look closely at the Excelsior just before she gets walloped by the Planar Shockwave: You can see the ship is trying to turn into the shockwave (which, incidentally, is what Sulu ordered the helmsman to do a few seconds later.) Evidently the helmsman was on the ball and did his best to minimize the initial impact since there didn't seem to be time to get out of the way.
More than minimizing the impact, the helmsman probably also realized that maneuvering in the shockwave and avoiding being dragged along with it required opposing thrust from Excelsior's impulse engines.
And no surprise that a damn good helmsman like Sulu would know how to pick a damn good helmsman for his own ship. They're cut from the same cloth.
The disastrous dinner scene is one which is explainable by a number of factors ranging from alcohol to neither side having a real interest in peace except for the Chancellor. However, it becomes Fridge Brilliance when you realize Valeris and Chang were both sabotaging it. The first by providing the Romulan Ale which is a Chekhov's Gun that allows them to frame Bones for incompetence or malice. The second is the fact Chang is a student of Earth culture to the point he knows who Hitler is and quoting him was bound to get a reaction from Kirk. A reaction which would insult Gorkon who would also get it.
The Klingon Blood Problem. It is a big plot point in this movie that Klingon blood is different from human blood when unmasking the assassin at the end of the story. The big assumption is that the two blood colors are different, i.e. the Klingons being shot earlier on are seen having pink blood and the assassin at the end is seen with red blood. Yet, the Klingons in all of the other Trek movies and TV shows bleed red. However... the difference isn't in color, and Klingon blood is normally red, like humans, only it turned pink due to some environmental mishap when the Klingon ship was under fire (loss of gravity? some gas tank got hit?). Notice how the person saying "This is not Klingon blood" when feeling Colonel West's spill is doing exactly that: feeling, not simply visually identifying the blood of the assassin. Klingon blood therefore must be of a different consistency than human blood, not color.
How did the Klingon warden know who was Kirk and who was the shapeshifter? Martia wasn't wearing her chains.
Kirk was a Starship Captain. When he says, "Not me you idiot, him!" he says it like he's giving orders, not like a punk ass criminal.
Not to mention that it didn't really matter who he shot first.
As has been noted elsewhere, Martia had slipped off her leg irons before leading Kirk and McCoy to the surface, while Kirk was still wearing his.
Martia-Kirk also has different eyes than real Kirk.
After Kirk dives onto the Federation President to save him from the assassin's fire, he curtly identifies himself: "Kirk. Enterprise." Why do this? Because the Federation President is blind — it's a subtle thing that's never explicitly stated in the movie, but upon re-watching it's surprisingly obvious. And naturally, the President would have no idea who just saved his life.
Take special note of when, just after the Enterprise crew beams in, all hell breaks loose and everyone scrambles, except the President, who remains at the podium looking around as if he has no idea what's happening since he can't see what's going on.
Also notice that before he looks at anything in the movie, he puts on a pair of glasses. . .a proto-VISOR?
It makes a bit of thematic sense too. If he's blind, he can't judge Klingons and Humans by outward appearances, and instead has to judge them by their actions and character. And the President is the main roadblock in the way of war between the Klingon Empire and the Federation of Planets after Chancellor Gorkon's death.
It hit this troper that this movie finally ties up the personal character storyline for Captain Kirk that was established in Wrath of Khan. In WoK, Kirk complains about feeling old, useless, and generally accepting the best times of his life are behind him, something that McCoy chastises him for, encouraging him to be like his younger self before he truly becomes an antique. Certain events bring Kirk to feel rejuvenated and youthful enough to once again take up the captaincy of a new Enterprise. When this Enterprise's time has come and Kirk is faced with his mortality once again in Undiscovered Country, Kirk plots a course to Neverland ("Second Star to the Right and Straight On 'Til Morning"), informing the audience and reassuring himself that he intends to never grow old. Something that Kirk apparently follows through with until his death.
A scene cut from Star Trek: Generations (because its poor quality made it completely unusable) would have would have had Kirk's first scene in that movie be a skydive from orbit!
The character arc actually starts with Star Trek: The Motion Picture: Admiral Kirk, Chief of Naval Operations, is chafing at his desk job and wants to get Back in the Saddle. Unfortunately, he hasn't been in command of a starship for several years, and the Enterprise isn't the same as it was when he commanded her, thanks to a major refit she underwent. He spends much of that film butting heads with the younger Captain Decker. So the first film has him realizing he's not the same young captain he used to be, the second film has him accepting his mortality, the third and fourth have him defying death by bringing Spock Back from the Dead and bringing the whales back from extinction to save all life on Earth, the fifth film was kind of a swing and a miss but should have been about how a mortal Kirk deals with the greater universe as a whole, as well as foreshadowing the shifting political tides away from what he knew for much of his life, and the sixth film is about letting go of life-long prejudices and accepting change.
Careful viewers will ask themselves "How did the assassins beams to Qo'noS One and beam back when Valeris was on the bridge?" Aside from settings the transporter to a timer, you'll notice there's a female enlisted officer in the transporter room when Gorkon and his party beam to the Enterprise. It's possible she had a peripheral role in the plot, not knowing why she had to beam two people over, or even thinking she was beaming over help.
Look at the energy surrounding the two when they beam. It's red. That means they were transported to and from Qo'noS One by the hidden Bird of Prey.
Not to mention a subtle embrace of his human heritage, tying in with his discussion with Valeris regarding Logic being the embodiment of wisdom or merely being the start of it.
Given that his human foster sister is later revealed to have suffered a fall from grace herself (going from a First Officer on the verge of getting her own command to being imprisoned for mutiny), the painting carries quite a bit of potential meaning for Spock.
Minor and paired with Fridge Humor, but after Kirk's fight with the giant alien on Rura Penthe, where he defeats him by kicking his knees, Martia tells Kirk that "not everyone keeps their genitals in the same place." Kirk asks if there's anything on the subject Martia wants to say, and she just grins. Of course, we find out later she's a shapeshifter... she can keep her genitals anywhere she wants!
In Star Trek: Discovery, General Chang has an interesting foil in the form of Kol. Both are ambitious Klingon leaders who hope to take advantage of new stealth technology to gain an advantage over the Federation in combat. But while Chang is a Cultured Badass with a healthy appreciation of his foes and their culture, Kol is brutish thug (with a dangerous cunning for Klingon politics). Chang and Kol are further contrasted by their deaths, at the hands of Starfleet officers who found a way to defeat their cloaking devices: While Chang has an Obi-Wan Moment, quoting an old human playwright, Kol instead spends his last moments screaming in impotent rage.
There's a Stealth Pun at play in McCoy helping Spock modify the torpedo: He's the only man that Spock would trust to help him perform Rocket Surgery.
Meta example: In this film, which was made after TNG had begun its run, Kirk and Spock take on the same roles as Riker and Picard, respectively: Kirk, like Riker, is the impulsive, manly cowboy officer who leaves the ship, gets in trouble and needs to be rescued, while Spock, like Picard, remains on the ship and begins the process of rescuing him in a strategic way. Spock is calm, contemplative and quotes classic literature frequently in this film, like Picard, and and he's actually the one who gives Kirk his mission, by volunteering him to be Gorkon's honor guard for the voyage to Earth.
Azetbur is named Chancellor in her father's stead. However, chronologically-later episodes of Star Trek: The Next Generation indicate that women aren't allowed to serve in the Klingon government. Continuity mistake? Or did something happen during Azetbur's regime that made the Klingons change it? Was Azetbur a particularly bad or incompetent Chancellor?
The Star Trek: The Lost Era novel "Serpents in the Ruins" offers an answer to this — Azetbur wasn't bad or incompetent, just that she continued her father's peace efforts and was killed by more a militaristic faction who believed the Empire had been at peace for too long and were no longer interested in accepting Federation aid. One of her successor's first acts was to simply ban women from serving on the Council, seemingly blaming her gender for her politics.
Note that this would also explain the above alteration to the timeline in "Yesterday's Enterprise" — a militaristic faction coming to power in the Klingon Empire, interested in ending peaceful relations with the Federation, would be impressed by a Federation starship — indeed, not just any Federation starship but the Federation starship Enterprise — sacrificing itself to save a Klingon colony and view them as worthy allies. Without that act, we end up with the war that brought the Federation to the brink of defeat.
For years, I never realized that the President in Star Trek VI: The Undiscovered Country was blind. Suddenly, the opaque pair of glasses he wears when looking at the Operation Retrieve plans make sense: they're an early form of the VISOR. Also, when Kirk dives over the podium and knocks him out of the way of assassin fire, he'd have no idea who did it — which is why Kirk says to him: "Kirk. Enterprise."
That one was actually more or less confirmed by Michael & Denise Okuda in the DVD text commentary for that movie — he was at least intended to be blind originally.