The Man Trap - S1-E2
Corrected entry: Despite the fact that McCoy has a tricorder that can scan for life forms, Kirk orders him to return to the Enterprise to scan for life forms there.
Corrected entry: Spock starts to lose control of his emotions and is desperately struggling with himself. He goes into a room and the automatic doors open for him. As they close behind him, Spock leans back against the doors. The doors should sense something approaching and open again, depositing our heroic Vulcan on his back in the corridor.
Correction: Throughout all Star Trek series, it is obvious that the computer monitors doors, and knows when to open them or not. For example, the doors of someone's quarters don't automatically open when a visitor arrives, the doors stay closed while the person 'rings the doorbell'. So, when someone leans back against a set of doors, the computer would keep them shut, to avoid the person falling backwards.
Corrected entry: Trelane says he studied Earth images that travelled to him at light speed, and earlier they establish Gothos is 900 years from Earth. But Trelane references Napoleon and Hamilton, who weren't around until 1800 or so. That would put this episode in 2700, but the original Trek episodes are set in the 2200's.
Correction: You assume that Trelane lived only on that planet. Rather, Trelane was a powerful energy-entity that did not actually live on the planet, he was just playing with it. Trelane's kind could go anywhere in the galaxy they wanted, at will. In fact, Trelane's parents merely allowed him to play with a whole planet to keep him occupied. As an energy-entity, Trelane had obviously observed Earth from much closer range, at some point, which is when he became fascinated with 18th Century Earth civilization and warfare. But, when the Enterprise encounters him, he's "playing" elsewhere in the galaxy, now 900 light years from Earth.
Corrected entry: Kirk says to the Admiral (actor Jose Ferrer) "Once more, Jose."
The City on the Edge of Forever - S1-E29
Corrected entry: After extinguishing the fire, Spock pulls a large vacuum tube out of the primitive computer power supply he's constructed on the bed. When the shot changes, the tube is back in place again. (00:30:00)
Correction: The tricorder can only scan the immediate area, but from the ship he can scan the entire planet.