Dr. Gillian Taylor's disappearance ended up on an episode of Unsolved Mysteries.Imagine it from the perspective of the people in 1986. Gillian Taylor shows up for work and discovers that the whales have been carted off to Alaska without her knowledge. She slaps her boss while calling him a son-of-a-bitch, then drives off in her pickup truck. Later, her truck is discovered in Golden Gate Park, and when she doesn't report to work later on, the police open up a missing person investigation. The media gets wind of the Cetacean Institute's assistant director going missing, and they put her face all over the evening news. This is where it gets weird. Gillian's photo is recognized by one of the witnesses at Mercy Hospital as one of the people who sprung the FBI's strange Russian prisoner and then vanished. Witnesses also say that she was assisted by two other guys, one of whom is later identified by others as a guy that Dr. Taylor was seen talking to at the Institute, and later having dinner with at the Italian restaurant. Investigators search for all possible leads, and they go through Dr. Taylor's personal life, but come up with nothing. So all of this, combined with the fact that her truck was discovered in a section of Golden Gate Park where people had seen strange goings-on, may have given Gillian's disappearance a mention on the Trek-verse's version of Unsolved Mysteries.
The Whale Probe was built by the Xel'Naga.The Xel'Naga discovered a primitive sentient species on Earth millions of years ago, who were the ancestors of today's whales, and sent the probe to check on their progress. After the fall of the Xel'Naga, the probe kept on monitoring Earth until it no longer recieved any transmissions. Due to a faulty FTL drive, it took two hundred years to get to Earth.
George and Gracie themselves are aliens, the Whale Probe is from their home planet and came to Earth because the crew went back in time and got them.]] George and Gracie themselves are aliens, and the "Whale Probe" was sent to determine where it was they actually went. When they lost communication in the past, they built the whale probe to figure out what exactly had happened to George and Gracie. For whatever reason, the whale probe took three hundred years to arrive to Earth. Had the Enterprise crew not have gone back in time and gotten them, the probe would never have arrived.
Dr. Gillian Taylor's disappearance did not affect the course of history.She told Kirk when the whales were removed from the aquarium that she had nothing without those whales. It's altogether possible that, in the absence of Kirk—or even, if word got out that George and Gracie were killed by poachers immediately after release—she would've committed suicide soon thereafter. So her disappearing from the timeline and reappearing in 2285 would not affect history.
The Klingon ambassador is spouting forth propaganda.Look at the video footage he has playing behind him. It includes footage taken from the Genesis promotional video from The Wrath of Khan with Kirk narrating where Carol Marcus ought to be. That looks a lot like they digitally altered the video to superimpose Kirk on top of Marcus. Also included is shots of the Enterprise self-destruct and the Klingons on its bridge- possibly taken from recovered flight recorder data, with Klingon text over it, as I recall. That looks to me like it's a deliberately doctored Klingon propaganda clip taken from what could be some Klingon news report, maybe. Combined with the Insane Troll Logic he spouts about Genesis being part of a plot to destroy the Klingon empire, which sounds a lot like propaganda from some crazy regime. Why? I imagine that some faction in the Klingon empire wants to set up Kirk as a renegade war criminal or terrorist in order to spare themselves the shame of dishonour, either of dying in battle for no good reason or pure personal gain, or of the subterfuge Kruge used in order to obtain the secrets of Genesis (which seems hardly an honourable course of action, being somewhat wily). This way, it looks like Kruge will be spared the shame of dishonourble defeat. (Mind you, there was if I recall some bit in the third film where Kirk's face was shown on the same video- which seems to undo this somehow. Yet how did that turn up if so?)
The whalers...Like the disappearance of Gillian, it's likely that their story will eventually find it's way into the media somehow- they've essentially just seen a UFO appear almost out of thin air. Another thought is that some of them were so freaked out by the experience that they gave up the business of hunting whales altogether, lest "the aliens" appear and get them for good next time...
Another UFO sighting occurred at NORAD or similar when the ship first came out of warp...Spock mentions that they'd probably already be visible to the tracking devices of the time. Imagine seeing something suddenly appear in orbit and then mysteriously vanish. Chances are amateur astronomers might have noticed a brief spot of light in the sky (real-life orbiting satellites, spacecraft, space stations etc. can sometimes be seen as points of light in the sky, reflecting the Sun's rays) too.
The pills McCoy gave the woman were created from the spores from "This Side of Paradise".While most people remember the mind-altering effects of the spores, a rewatch of the episode also shows that they caused those affected to regrow lost tissue. After having discovered this through personal experience, McCoy thought that it would be awesome to harness that power. After much experimentation, he created a pill that could regrow organs without the mind-altering effects.