I think there was an episode in ST:TNG where a ship was attacking the Enterprise with lasers and Picard and Riker were astonished wondering how they didn’t realize how useless those were against the Enterprise. The laser ship could shoot forever and never do damage to them.
Duh, it’s right in the name – phasers – they phase you to another existence. Jeez.
The Khan-blood thing is put to hilarious use in the How It Should Have Ended episode of that show.
Subspace. It comes from subspace.
Star Trek: The Next Generation had one episode mentioning Betazoid pon farr. The woman spent a lot of time and effort trying, unsuccessfully, to seduce Picard over candlelit dinners and wine, or something like that. How hard can it be to get laid in a ship full of sailors?
That would have been Lwaxana Troi, Deanna’s mother, who was played by Majel Barrett Rodenberry; she was well-known for her dalliances, flirtations, and many husbands (and was, essentially, somewhere between Auntie Mame and Zsa Zsa Gabor).
In the episode “Manhunt,” she was, indeed, going through “the Phase,” and had specifically focused on Picard as the object of her affections.
As far as Episode 0 itself goes, it seems kind of baffling that the Federation never held negotiations with the Talosians to learn more about their mind control capabilities. Imagine developing a device that would fuck with the other guys’ heads. You just aim the beam at the bad guys, set to “terrified”, and they become unable to fight. No blood, no destruction, just simple crippling victory. The machines used in Dagger of the Mind, The Squire of Gothos, Who Mourns for Adonais, Catspaw, That Which Survives and The Savage Curtain might also have been adapted for similar defensive use. For a Starship on a voyage of discovery, their blinkered incuriousity about particular things is kind of hard to fathom.
Fast-froward to the lamentable final episode of TOS: here we have a machine that can transplant consciousness from one person to another. A similar theme appears in Return to Tomorrow and is kind of flirted with in a few other episodes, but is never really explored. Imagine how much more effective it would be to just get into someone else’s head instead of using that little flip-phone thingy. That guy in the landing party would not have to describe the situation to the guy on the ship if the guy on the ship could see it with his own (the other guy’s) eyes.
Those glowly half-globe things on the front of the nacelles are Bussard collectors. Antimatter can be found in small quantities orbiting gas giants, and these things apparently vacuum them into some containment chamber inside. [/fanwank]
I mean, that one does make sense within the setting. The Federation is a post-scarcity society because they can convert matter into energy and back at will.
But it is in the sense that, in a real universe where such is easily accomplished, those time cop people at the Temporal Institute would be CONSTANTLY trying to unravel and repair damaged timelines. In-universe the time travel Pandora’s Box has already been opened (by at least two means incl. the Guardian of Forever), and there will ALWAYS be evil or short-sighted people fucking with the past.
You would think that the TOS enterprise would carry some extra Dilithium crystals, eh?
“Is this a Kirk case? I hate Kirk cases.”
Which, apparently, they are. But a point remains that all they have to do is screw up once… and they may poof out of existence themselves. Which actually happens in the course of Enterprise, but for… reasons… the main character isn’t wiped out by the paradox, even though everyone else is. In a vaguely more rational scenario, those who use time travel frequently must have some sort of shielding so they can go back and fix things with of course, some degree of drama inducting duration and limited numbers who can be protected.
Or they borrow Quantum Leap’s Bakula, and he’s always busy fixing things while himself untethered from the timestream. 
Ah Star Trek, 'tis a silly place.
a.k.a. Captain Archer. 
Yeah, there were some very unsubtle moments in the Temporal Cold War stories with Bakula/Archer…
Sometimes the wink, wink, nudge nudge semi-subtlety was abandoned for a sledgehammer.
There were the “subcutaneous transponders” from “Patterns of Force” (The One With The Nazi Planet) to enable the ship to track people if they got separated from their communicators, which seems like something that would be handy in pretty much any landing-party situation.
Relevant TV Tropes topic.
Star Trek live action TV, movie, animation, and literature instances of the phenomenon are plentiful but y’all have already covered many of them.
Seventeen separate temporal violations. The biggest file on record.
I’m not sure this really counts, because it’s something that (as far as I know) they didn’t actually discover…
Transporters as weapons. There are any number of times that they figured out how to get the transporter to work through shields by modulating the harmonic frequency of a reverse tachyon pulse in the main deflector, or whatever.
Why not use the transporter as a weapon? Just start transporting away important parts of the enemy vessel - such as tiny but important pieces of the main reactor, or key structural components, or the enemy captain’s aorta. It doesn’t even matter if you can’t rematerialise them, as long as you can do the dematerialising part.
Or that magical elixir in “The Deadly Years” that not only cures radiation poisoning but miraculously restores youth and leaves one in perfect health.
Hell, just beam a cobalt bomb onto the enemy ship and blow it to smithereens.
One thing that got me was Scotty beaming down Ambassador Fox while the shields were up in “A Taste of Armageddon” and Sulu being unable to beam up Kirk and Spock in “Errand of Mercy” because he can’t lower the shields.
Best not to dwell on such details, I think.