Spock tells Zarabeth he came from a world "millions of light-years away." The galaxy is only about 100,000 light-years away - the area the Federation has explored even smaller
When Spock and McCoy first
arrive in the frozen tundra, Spock attempts to use his phaser. It
doesn't work and Spock blames the temperature. Yet, in "The
Enemy Within," Sulu uses his phaser in 70 below zero temperatures.