On the bridge, Sulu breaks his right leg, but in sick bay, his right leg splint has moved over to his left leg.
At 2:21, McCoy suddenly appears on the bridge next to Kirk. Several previous shots include a view of the turbolift doors. None of them show the doors opening for McCoy's entrance.
A normal-sized human has about the same density as water, which is why normal-sized humans are neutrally buoyant in water (more or less). When shrunken Nurse Chapel falls into the aquarium, she sinks like a stone. This is correct because her size is smaller but her mass has not changed. This would mean she is now much more dense than she had been. She would not have been able to swim back to the surface of the water to call for help. She would have drowned before anyone could have helped her. The water would not have changed density because water is not organic, and the thread rescuing her (inorganic as it doesn't shrink) - as well as the side of the aquarium - would have to withstand her full weight.
Spock says they would shrink by a factor of 32.9, and Kirk replies they'd end up being 1/16th inch. The correct height for that factor would be approximately 2 inches, for a 6 foot tall person.
Early in the story, McCoy explains the shrinking of the crew as the spacing between molecules decreasing, which is why the crew have the same mass as before. They're just shorter. Later, however, it takes three engineers to operate the transporter levers. If they have the same mass (and weight), a single person should still be able to push buttons and move levers.
At 6:27, a gold-shirted crewman appears in the mess hall without his comm-badge. In effect, he's drawn out of uniform.
At 2:27, McCoy stands next to Kirk. Uhura sits in the background behind McCoy, but her left arm and legs appear in front of McCoy's shoulder, making her look half his size and riding his back.
Spock picks up the dilithium crystal pieces as if they were one merged chunk.
A few times in the planetary backgrounds, volcanoes are shown erupting. More than a few times the eruptions and fire are not moving. It's a totally motion free background.
Back to normal size, Kirk stands straight as he operates helm controls, making the console seem designed for operation while standing rather than while sitting.
Christine's first "help" is heard when her head is underwater, yet is spoken clearly as if not underwater. Afterwards, she says "help" several more times when her mouth can be seen not moving.
Only a few inches tall, Kirk and Arex take Sulu to Sickbay. How did they (and why would they) haul Sulu, broken leg and all, up to the surface of an examination table for tiny Dr. McCoy to examine? The effort would waste time; they should have all remained on the floor.
The fate of the sensitive animals is not explained.
Spock says the coral in the aquarium is also shrinking, yet when Kirk rescues Christine from drowning, the height of the small coral towers over Kirk.
When the bridge crew (per se) and a pile of dilithium crystals are beamed aboard, the dilithium pile appears without a transporter pad beneath it. (It's also good that Christine abandoned the idea of wearing her bracelet as a necklace, otherwise she's be strangled upon her sudden size restoration in her beam back to the ship.)
Sulu and Arex construct two makeshift ladders to reach helm controls. The ladders appear to be wood lashed with something like thread (seen in close-up at 11:15). Wood, being organic, should shrink along with Sulu and Arex. They've effectually built ladders that would soon be no good. In addition, where on a starship would these guys lay hands to rough lengths of wood to construct ladders, especially on the bridge? The makeshift rails alone would start off in lengths of 20'.
Kirk tries to establish contact with the planet inhabitants by threatening them, destroying a nearby outcropping of dilithium crystals as a demonstration of his destructive and vengeful capabilities. This is not like him and against Starfleet directives.
Opening up communication, Kirk gives the inhabitants of Cephas 1 just one minute before he hits their minuscule city with a full barrage. This is either an empty threat or a callous disregard for other life, other life forms, and the lives or well-being of his abducted bridge crew. No explanation for his sudden abandonment of the ship's first priority: To explore strange new worlds and seek out new life and new civilizations.
In desperation and without reason, Sulu wants to shoot the planet with ship's phasers. Hysteria and panic are not among his characteristics.
Subtitles spell the star as Cepheus, but the characters pronounce it as Cephas. Since both are legitimate Greek names, either is possible.
After beaming his bridge crew (per se) aboard, Kirk orders all ship's personnel to report to the transporter room for beam down. This is a pretty demanding, fairly callous command. By this time, the crew is too small to either reach controls, trip electric eyes of the automatic doors, or avoid getting stepped on by Captain Kirk. All efforts to reach the transporter room, singularly or in groups, is a dangerous venture for the miniaturized crew.