When Spock is playing chess, he makes a move before McCoy walks in. He ignores the computer's spoken move (bishop 1/2 level right). He then makes another move, and the computer repeats its previous move.
When Kirk is addressing the members of the Court on the bridge, he said that he had shut down the engines. However, in the remastered version, the shot of the Enterprise shows that the engines are still running.
Although Kirk has no objection to his ex-girlfriend serving as his prosecutor during the court martial, there is no way in a system of justice based on precedent, the rule of law, and individual rights that she would ever be allowed to serve as the prosecutor for his case due to an obvious conflict of interest.
Kirk's control panel is of an absurd design where buttons which do significant things like send ship to red alert or jettison the pod lack any kind of flip open covers or even side bolsters to prevent being inadvertently pressed. No craft would be designed in this way.
There are four officers sitting on the court martial board. However, all courts martial always have an odd number of officers sitting so there would be no tied vote. While things may change in the future, this almost certainly will never change.
Kirk's lawyer, Samuel T. Cogley, is unprepared for Kirk's case in the courtroom. Cogley has no files or folders of paperwork that would indicate he has prepared a case. All he brought into the courtroom is one legal pad and five old textbooks.
The idea of recording and filtering out the sounds of the heart beats of everyone on board is ridiculous. The rate and intensity of heart beats vary with time (especially in a stressful situation like this) and would no longer correspond to the recording. The Enterprise has more sophisticated means of detecting life-forms than the sounds of heart beats or breathing.
A spacecraft in a stable orbit does not require engine power. It would only drop out of orbit through atmospheric drag or the slowing down of its orbital speed. However, it is frequently established elsewhere in Star Trek that something as massive as a starship cannot maintain a stable orbit indefinitely with power.
It's doubtful that only 3 out of 430 people aboard the Enterprise would know how to reprogram the computer to be able to win at chess. The ship is staffed with all kinds of specialists. There would be a great deal with computer expert knowledge to do it. However, when Spock identified the people who would "be able" to make the changes, he identified, Kirk, himself, and the "records officer," which position is acknowledged to be vacant at the moment. Based on this description, it is likely a matter of authorization to access and modify the record logs rather than ability to make the changes.
After Spock wins his fourth game, he calls the transporter room to let them know that he and Dr. McCoy are beaming down. At that point, why did he not contact Kirk and/or his lawyer (Cogley) to let them know that they had new information critical to the case - before the trial resumed? It is likely that no communications from outside were allowed in the courtroom as this kind of thing is fairly common.
Kirk's defense lawyer (Cogley) is permitted to call Captain Kirk to the stand, even though the prosecution had not rested its case. One is left to assume that Kirk was possibly next in line to be called by the prosecution anyway, but the fact that the defense was permitted to question Kirk first indicates that the case had changed to the defense phase, again, without the prosecution resting its case. However, this is only a certainty in the American court system. In the Star Trek universe, there are those court systems that allow both prosecution and defense to present their arguments at the same time.
Kirk's lawyer brings a bunch of reporters (bound compilations of written opinions) with him to prepare for the case. this makes no sense for 2 reasons. As Kirk notes, the computer can store all the reported cases (which are thousands of volumes). The lawyer nonsensically suggests the books have more or different information than the computer, but the computer would have the exact same opinions (just on a screen instead of printed page, and with all reporter volumes accessible not just the dozen or so (out of thousands) that the lawyer brought to Kirk's office. Still, the computer screen cannot bring the tactile sense of holding a book in one's hand.
As part of the experiment to prove Finney is alive, McCoy uses a device to eliminate the heartbeats of the personnel on the bridge one by one. However, Kirk then has Spock eliminate the transporter officer's heartbeat remotely. By that logic, Spock could have simply eliminated the heartbeats of the bridge occupants all at once and saved time.
While the ship's audio sensor is sensitive enough to pick-up the sounds of the crew member's heartbeats, it does not amplify the sound of Kirk and McCoy's voice as they explain the process.
When he goes into the room where Finney is hiding, Kirk goes between two pieces of equipment. As he comes to the other side, a shadow of the camera is seen moving at the upper left of the screen.
During the prosecution's questioning of witnesses, she is clearly leading the witnesses on the stand. The defense should've called "objection" on her multiple times. It's unlikely this kind of practice would be acceptable in a famously fair and enlightened society like the Federation.
Captain Kirk and his lawyer should not have been surprised to see what the (doctored) computer record shows about his actions on the bridge when it is played for the court. The fundamental process of discovery which would be followed in any legitimate court proceeding requires the prosecution to turn over any evidence to the defense, particularly the one crucial piece of evidence underlying their case against the defendant, to allow the defense to prepare its case. It would be highly unlikely that the defendant and his lawyer would be seeing the record for the first time during the court proceeding itself.
The computer is supposedly amplifying all of the sounds on the ship. If everyone is hearing the heartbeats of every person on the ship, they should also be hearing the breathing of every person on the ship, which is significantly louder.
At the beginning of the episode, there are clearly two starships visible in orbit around the planet where the Starbase is located. It would be unusual for the Starbase to be devoid of other starships, yet when the Enterprise's orbit begins to decay, no call is made to request another ship to tractor the Enterprise away from danger.
The Court expressed concern that with the engines shut down, having the engine room crew beam down could be a problem as the engines might not be restarted before the orbit decayed completely. However, the engine room crew did not have to beam back up to restart the engines, because Spock and the helmsman were able to restart the engines from the bridge by themselves.
When Kirk is about to give his demonstration on the bridge he states that the ship's computer has audio sensors that they can boost by "1 to the 4th power". 1 to any power is only 1 so there would be no boost at all. He probably meant to say 10 to the 4th power, which is 10,000.
When McCoy is eliminating heartbeats, he positions the device to the center of Mr. Spock's chest. McCoy well knows that Vulcans do not have their heart located at that position, which is a running gag in TOS.
During the trial, only Spock and Kirk are identified by name and serial number by the computer when they take the stand. The Personnel Officer and Dr. McCoy are not - the computer starts with the witness's rank and current station. This is out of 'character' for the computer, as each person's personnel record would properly identify the person by name and rank at the beginning.
In the teaser, Jame attacks Kirk and screams that he's a murderer for killing her father. In a later scene, she has done a complete reversal in her attitude towards him. A scene in which she forgives him was cut from the final episode, which results in Jame completely changing her mind for no visible reason.
When Shaw questions McCoy, she states that he is an expert on psychology. When she presents him with the scenario that Kirk would come to hate Finney simply because Finney hates Kirk, McCoy goes along with it. He should dismiss that scenario as impossible, and point out that his job as the Enterprise's Chief Medical Officer not only requires him to be an expert in psychology (which Shaw has already entered into the record, thus opening the door for his response) but in particular an expert on the psychology of the ship's captain. If McCoy lacks the legal finesse to make that counter-argument, Cogley should.