When the Megans attack the Enterprise, McCoy is seen wearing a yellow command uniform during one shot.
When Lucien first transports Kirk, Spock, and McCoy to Megas-tu, Kirk asks McCoy if he's alright. McCoy says he's OK, but he could give a more definitive answer if he had a medical tricorder. He then moves his arms, and a shoulder strap from a medical tricorder can be seen on McCoy's left shoulder.
Around 7:07, after Lucian first takes Kirk, Spock and McCoy off the Enterprise, McCoy is drawn without bags under his eyes, making him look younger and quite different.
At 19:32, the camera pans the Enterprise crew locked in the stocks, including Spock and Capt. Kirk - but Spock and Kirk were, by this time, released from the stocks to serve as defense attorney and the second witness, respectively.
The star date given in Kirk's log is 1254.4. This places it before the original Star Trek (1966) took place, when McCoy, Arex and M'Ress were not on the Enterprise, and several other crew rollovers had not yet happened either. Surely the TAS writers did not intend this story to be a prequel.
The Enterprise travels to the center of the galaxy, not the universe, to seek the center of the big bang.
The ship's life support systems give out, and the bridge crew immediately suffers oxygen deprivation. Given the size of the bridge, it should take longer for the oxygen to run out.
The people of Megas say they saw their friends burned in Salem. This is a popular misconception. None of the people found guilty of witchcraft in Salem were burned. They were hanged or pressed to death.
When Spock says he is not of Earth and will speak on the behalf of the Enterprise crew, Lt. Arex should have spoken up instead. Half of Spock is from Earth; all of Arex isn't.