Factual error: 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.
Revealing mistake: Chekov covers Mara's mouth with his hand when he assaults her in the corridor. Kirk intervenes and throws Chekov against the opposite wall. Brown streaks form where Chekov's hands hit the wall, most likely from the brown makeup used to create Mara's skin tone.