Corrected entry: Kirk says the Platonians moved to their current planet when their last planet "novaed." Planets don't go nova - stars do.
Corrected entry: Kirk's line, "We're a doomed ship, traveling forever between galaxies" is the only time original "Trek" committed the scientific blunder of confusing galaxies and solar systems. The Enterprise wasn't capable of intergalactic travel (that's leaving one galaxy and reaching another). Yes, it strayed briefly out of our galaxy several times. But it did not - and could not - cross to another one. That would take a warp 100-plus drive and thousands of years. (00:37:25)
Correction: In the course of a normal human lifespan, Enterprise wouldn't be able to travel to another galaxy. Given enough time however, it certainly could reach other galaxies. The plot of this episode is that they were (as you stated), doomed to travel forever (as opposed to the rest of their lives). Given all eternity, they would be traveling between galaxies.
Corrected entry: Mira is supposedly weightless in the pressure chamber. But her tunic skirt remains perfectly flat and her long hair also "forgets" to float - it's hanging straight down below her head. Selective anti-gravity? (00:44:50)
Correction: If she were lying flat on a surface, with her hair hanging down, when the anti gravity was activated, the cloth and hair would remain in this original position due to the absence of any gravity to affect their position. There are many videos of objects in weightlessness where astronauts placing an object in a spot in front of them, and it staying put, unless it were given motion from the touch of the astronaut.
Corrected entry: Garth, shape-shifted as Spock, applies the Vulcan nerve pinch to Marta. Since Garth can only imitate the physical appearance of someone and not that person's life experience, Garth would have no knowledge of the Vulcan nerve pinch. The same could be said of when Garth shape shifts into Kirk - if Garth had Kirk's knowledge, he would have known the counter sign that Scotty used to challenge him for the beam up.
Correction: Garth, prior to being able to shape-shift and going insane, was a well respected Starfleet Captain and skilled tactician. In fact, he was required reading at the academy. Captain Garth would no doubt have extensive knowledge of Vulcans and the Vulcan nerve pinch, including learning how to do it.
Correction: He probably meant that their previous planet orbited a star that went nova, ie the planet was "novaed".