Character mistake: Spock tells Zarabeth he came from a world "millions of light-years away." The galaxy is only about 100,000 light-years across - the area the Federation has explored even smaller.
Character mistake: Elaan says "If I have to stay here for ten light-years..." A light year is a measure of distance. (00:24:03)
Character mistake: Watkins is asked to check the bypass valve, there the woman appears, asked about the panel, it's obvious she's not part of the crew. Instead of calling security or Mr Scott he starts to describe the panel, which I'm assuming would be classified information.
Answer: Kirk was getting his physical and Dr. McCoy probably turned off communications, because if he hadn't, Kirk would have left and headed straight for the bridge, leaving McCoy irritated.