As Kirk beams up after baiting the creature, his arms are raised and he is using his communicator with his left hand. As he materializes on the Enterprise, his arms are lowered and he is no longer holding his communicator.
As Kirk converses with Ensign Garrovick in the ship's corridor, in their respective close-shots the curvature of the wall behind Kirk matches that of the wall behind Garrovick (who is directly facing Kirk); the corridor curvatures should be opposite for each character.
Just before Spock phasers off a sample of rock, he is standing directly behind Kirk's right shoulder. As he announces his intention to fire the phaser, he is suddenly several feet to Kirk's left, and his phaser hits the rock from that direction. There was no time for him to have changed position.
Lead changing into gold is not a change in molecular structure. It is a change in atomic structure.
Ensign Garrovick states that less than one ounce of antimatter is more powerful than 10,000 cobalt bombs but the actual energy yield is only about 1.2 megatons TNT. The implication that a cobalt bomb releases more energy than a "regular" nuclear weapon is incorrect; a cobalt bomb is just an ordinary thermonuclear weapon "salted" with cobalt to produce long-lived Co-60 in its fallout.
The Cloud Creature is depicted entering the Enterprise through a jammed impulse engine vent, opened to the vacuum of space. As it is also depicted as unable to get through solid material, the only way it could have reached the interior would have been to open the duct or manifold corresponding to the jammed impulse engine vent to the ship's interior. This act would have depressurized the Enterprise, killing all aboard.
If the rock at the beginning of the episode was as hard as Kirk said, Spock's phaser should not have been able to penetrate it. They would have needed mining equipment for that.
The preceding statement is inaccurate. The hardness of materials is completely unrelated to the ability of energy-based weapons/tools to penetrate them. (Lasers, for example, are used to etch diamonds.)
McCoy's short-sleeved medical tunic bears an engineering insignia instead of the usual sciences division insignia for the entire episode.
When Kirk and Spock find a large rock made of the super-hard material they are looking for (20 x harder than diamond), Kirk hits it and it makes a 'clonk' sound as if it is made from hollow styrofoam, which of course it is.
In the first scene, Spock phasers off a sample of rock, which has been clearly pre-cut to fall off on command.
When Garrovick fires his phaser at the cloud creature, the blast is a standard phaser beam, even though Kirk ordered the men to set their phasers on a disruptor setting. In A Taste of Armageddon (1967), the disruptors were sonic weapons which did not generate any visual effect, but did create an audible sound blast. In Wink of an Eye (1968), a similar-sounding weapon was used by the Scalosians - but it created a visible effect.
When Kirk is re-materializing in the transporter after baiting the creature a black spot appears on his face momentarily during the process.
Technically, Kirk should not be surprised that the Enterprise phasers passed right through the gaseous cloud. It is not a tangible entity. A good comparison might be a simple dust cloud. Throwing anything at it will pass right through it. It also can't be physically touched. However, why the photon torpedo were able to detonate upon contact rather than pass through is unknown.
Kirk orders Scotty to flush radioactive waste into the ventilation system to force the cloud creature out of the ventilation system. The ventilation system would spread the radioactive waste throughout the ship, radioactive waste would spew out of the vents and the radiation and toxic chemicals would kill everyone aboard the ship.
The plot complication involves perishable vaccines needed for a Federation colony, which the Enterprise is supposed to receive from the USS Yorktown. It is never explained why the Yorktown can't simply deliver the vaccines itself.
When the creature invades Garrovick's cabin, Scotty is on the Bridge. However, when he pages Kirk a couple of minutes later, he does so from Engineering, and is off the Bridge when Kirk and Spock return to it. No explanation is given for why he would leave the Bridge at that point, particularly when he is in command.
The security officer, Ensign Garrovick, is ordered to the bridge. When he shows up, Kirk doesn't know who he is, nor that he's the new security officer. For someone in Garrovick's position, the captain should know who he is. And in this case, that he is someone related to someone Kirk used to know.
At the hearing, Ensign Garrovick describes the creature as being "several cubic meters" in size, but his distance as "about twenty feet away," mixing two different systems of measurement. It's unlikely that people in the Trek Universe would have any use for the antiquated system of feet and miles.
Garrovick appears to be in charge of the other security officers, yet he has no command rank, and is only an ensign.
When Kirk beams down to the planet with 5 security men, he orders Garrovick to take 2 of his men and circle around to the left. Garrick and two men then go off to their right.