While in Kirk's body, Lester makes a Captain's Log explaining her plan.
This is one of the few episodes of the series where Nichelle Nichols (Lt. Uhura) does not appear at all. Barbara Baldavin, listed as a guest star, appears as the ship's Communications Officer.
At the climax of the episode, when Kirk-as-Lester is fighting with the Doctor and Lester-as-Kirk is screaming, "Kill him! Kill him!", Scotty, Bones and Spock just casually stroll out of the cell to watch the action, instead of rushing into the struggle. Note also the lack of any security guards on the cell of mutineers.
A slightly odd moment. When Spock rescues Kirk-as-Lester from the cell and knocks out the guard, notice he takes Janice/Kirk's hand to lead her away, even though he knows it's really the captain. We've never seen Kirk and Spock holding hands before, and it seems unlikely Spock would feel more emotional/compassionate just because Kirk is now a woman.
Why doesn't Kirk-as-Lester just slip out of the loose restraining strap that is holding him down in sickbay? Why does he waste time cutting himself out?
It seems that when the hearing is taking place, Spock might be able to offer some more proof that the captain is not the captain by asking him to recall specific memories that only Kirk would know. He could also ask Kirk-as-Lester to recall specific stories and memories about the Enterprise. Instead, the only proof Spock offers is his mind meld. Similarly, Kirk-as-Lester doesn't attempt to establish his identity by revealing information only he and Spock or McCoy could know. Instead he talks only about things in the "public record".
How could Kirk order an execution? In the episode "Mirror, Mirror", Spock said to the Kirk from the alternate reality that his authority on the ship is extremely limited when Kirk said that he would execute Spock. Besides, Sulu and Chekov claim that executions are illegal. Don't the security guards care about Starfleet regulations? Why don't they turn on Kirk when he starts to demand executions?
Kirk in Lester's body strolls off away from Spock toward a solid wall when he calls for a vote. He's supposed to be stepping outside - according to the book Star Trek Lives the director screwed it up despite Shatner trying to explain to him there wasn't a door there.
When Lester-as-Kirk sits in the captain's chair, Chekov vanishes/reappears in different shots of the bridge.
When they first arrive, McCoy examines the unconscious supposedly stricken Lester - she's faking but he doesn't notice a thing despite examining her with his medical wand.
Chekov says only General Order 4 carries a death penalty - he's presumably referring to the ban on travel to Talos (from "The Menagerie"), but that was General Order 7.
Spock claims complete entity transfer has never been done anywhere in the galaxy - they did it in "Return to Tomorrow" with Sargon and the others.
McCoy administers a psychological test to Lester-as-Kirk that reveals their basic emotional structure - in court he testifies it matches what Kirk had when he took command of the Enterprise. Lester, who is portrayed as nutty as a fruitcake, would hardly have the same emotional state as Kirk (at least, you'd hope she would not!).
How is Lester-as-Kirk going to get Kirk's safe open? She doesn't have his memories of the combination or anything and it uses a number sequence to open.
Lt. Galloway was the character killed by Captain Tracey in "The Omega Glory" and yet here he is! Maybe he had a twin brother.
The mechanics of the portal are unclear. Kirk, Spock, and McCoy are able to communicate across two different time eras, suggesting that the portal is open to both eras simultaneously. There's no reason for that to be the case, and it seem unworkable: if the portal is open to both eras simultaneously, where would something going through end up? What if people entered their portals from two different eras simultaneously? Also, both portals remain open at least long enough for Kirk to find his way back. This may mean Atoz left them open once the landing party realized they needed to be "prepared" and come back to do so. However, he's not there when Kirk returns, waiting to prepare him. What else does Atoz have to do?
Since Atoz knows that the landing party is unprepared, it's not clear why he doesn't call out and help them find their portals so they can come back and be prepared. He has no reason to believe that they want to do anything else, and until the very end seems to think they're natives.
Atoz seems deliberately difficult. The disc that McCoy watched was one of only two that were being used. McCoy left it in the viewer. Even if he put it away for some reason, how hard can it be for him to remember where he put it? Instead he and Kirk spend time going through every disc relating to the era.
Atoz tries to push Kirk into the portal and send him to another time. However, as established earlier if a time traveller isn't prepared then they'll die. It's also established that if a time traveller is prepared for a historical era, they can't survive in the present. Since Atoz is trying to save Kirk's life, he must have used the Atavachron to "prepare" the captain for whatever era he's sending him too. But... there's no indication and no time available where Atoz can or does unprepare the captain so he can survive in the present.
It's not clear why the Enterprise visits Sarpeidon. The Prime Directive would presumably keep them from interfering, and even if it wouldn't they don't have the resources to evacuate an entire planet, much less in the three hours between when they arrive and when the sun goes nova. Apparently they're there just to see why everyone disappeared... three hours before the population would have disappeared for good anyway. And then turn around and do nothing if they do find any survivors.