I guess a lot of these stories were much like the Twilight Zone. In fact some were taken from the Twilight Zone. 1960's Earth and the 50's were post war. People like Rod and Gene and obviously others saw, if we do not learn to understand one another we would end up blowing ourselves up.
Back to your selection, The Empath, so there was still hope that maybe if a planet was doomed (for what ever reason) an Alien force might step in and try to save some planet if they could.
As to the Big Three: (different episodes)
Dr. McCoy: Shut up, Spock, we're rescuing you!
Mr. Spock: Why, thank you, *Captain* McCoy.
Kirk: "Don't you think you better check with me first?"
Spock: "Captain?!?! JIM!!!"
Actually for my third example is the second movie. Spock's death scene with the shield between Kirk and Spock is quite powerful.