Robert Walker was 26 years old when he played 17-year-old Charlie Evans in "Charlie X".
Michael J. Pollard (27 years old) and Kim Darby (19) play pre-pubescent children in "Miri".
Fake Nationality: William Shatner and James Doohan (both Canadians) play an American and an Scotsman, respectively. Walter Koenig is a partial case: his parents were Russian Jews but Koenig himself was an American citizen playing the Russian Chekov. Nichelle Nichols (an American) played Uhura, whose native language is established as Swahili—implying Uhura is from somewhere in eastern Africa. Worst of all—Ricardo Montalban, a Mexican, played Khan Noonien Singh!
The show is often credited as the inspiration for cell phones, but it also accurately predicted the tablet PC.
The military and many high-level police agencies are experimenting with non-lethal heat and sound beams to disperse riots and disarm attackers without killing them. Phasers On Stun anyone?
No Budget: In order to cut costs, incidental music avoided scoring anything for violins. Melodies in the strings are played by violas. Violinists charge that much more, apparently.
The Other Darrin: Shatner's predecessor, Jeffery Hunter, played Captain Pike in "The Cage". This footage was later re-used in "The Menagerie", with Pike himself appearing a motionless deformity in an iron lung-type device. This was primarily to disguise the fact that Hunter was now a film star and thus unavailable; this new Pike was played by a lookalike (such as he is) named Sean Kenney. What's interesting is that Captain Pike was retconned into Kirk's predecessor, as well; He was the original Captain of the Enterprise, with Mr. Spock as his first officer. This is still canon in the Abrams film, in which Bruce Greenwood plays Pike.
Red Shirt: Although the Trope Namer, the first red-shirted casualty doesn't appear in series until episode 7 ("What Are Little Girls Made Of?"); the very first casualties are blue-shirted Science Team and gold-shirted Command squaddies.
The Rural Purge: Inverted - it should be noted that the demographic information that led to the Rural Purge wasn't available from Neilsen before 1970 or so; had it been, Trek would've benefited since it attracted advertisers' favored demographics.
Throw It In: According to legend, the stagehands didn't like Shatner very much, so in the episode "The Trouble With Tribbles," they continued the avalanche of tribbles onto the actor much longer than the script called for. (This includes the final Tribble which bonks him on the head at the end.) Shatner can clearly be seen glancing up at the prop men with annoyance.
Actually, the producers had Nichols read Spock's part in her audition because the Uhura part hadn't been written yet or even named. She had them describe Spock's character to her, instead of Uhura, "so you can see whether or not I can act."