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.
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.
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.
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!
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.
Azetbur is named Chancellor in her father's stead. However, chronologically-later episodes of Star Trek TNG 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.
When the Enterprise's crew goes rummaging through Klingon phrasebooks to respond to an incoming hail, why could they not just configure their ship's Universal Translator to provide a one-way translation of the incoming hail? And could they not configure it to provide an in-ship text translation of their response and recite that across the com channel?
Because Nick Meyer wants his books, dammit!
Handwaved in the movie by having Scotty mention that the Universal Translator would be recognized, so Uhura had to speak Klingonese in order to fool the listening outpost into thinking that they were a Klingon ship. Nichelle Nichols had protested this scene (saying that Uhura would have been fluent in the language of their main enemy) but was overruled.
In a strange example of Enforced Method Acting done by the actor and not the director, Nichelle Nichols' apparent disgust upon closing the channel seems directed at Nick Meyer, rather than the Morska Listening Post.