Some quick thoughts:
Is everybody supposed to look as sweaty as they do, or is that a result of the HD remastering process?
Considering he was only in his 30s at the time, Leonard Nimoy already looked shockingly ancient.
Surprised that no attempt really was made at an "origin story," as would be found in later Trek series, where crew members are assembled and the ship is introduced as embarking on its maiden voyage. Instead, we begin right in the middle of it, with the characters seemingly having been at this for a while already. It makes sense, I suppose, that the as yet unproven series did not have the luxury of time to spend on such formalities before digging into the action. I understand "The Man Trap" was not the first episode produced, nor the first chronologically, but was chosen to air first precisely because it was deemed more exciting.
That said, the actual action sequences are close to B movie-level in their execution. When the alien menace turns its attentions to Kirk, Kirk stands paralyzed and helpless, but it wasn't clear to me if this was 1) because of some theretofore unmentioned numbing power of the alien (beyond its disarming ability to take on the appearance of a loved one, which was clearly not in play against the captain), 2) because Kirk was struck dumb in fear, or 3) because the awful choreography and sense of timing of the direction, coupled with William Shatner's perpetually posturing acting style, only made it look like he was frozen mid-pose (and, note, when the camera cuts away from him and then back, he's not even in the same frozen pose but a slightly different one!), when, really, everything was supposed to be happening much more quickly and naturally than it was being performed.
That last possibility seems unlikely, until next in the scene, when Spock arrives to try to stop the alien, which is posing as Nancy Crater, a woman from McCoy's past, whom the doctor still loves. To the hesitating McCoy, who is unable to turn his phaser on the creature in Nancy's form, Spock yells, "It's not Nancy! If it were Nancy, could she take THIS!" as the Vulcan delivers furious yet ineffective two-fisted hammer blows left and right upon the Nancy creature's face. What's odd is that Spock begins speaking before his first blow even lands, suggesting that he already knew it would be useless, even though there had been nothing prior to suggest that the alien had any such superhuman strength.
What redeems the episode are the more contemplative dialogues between Kirk and Professor Crater, who has taken to protecting the creature, despite it having claimed his own wife as one of its victims. Crater, reasoning that the alien, the last of its kind, was only using its ability "the way we would use our muscles and teeth if necessary, to stay alive," can bear it no ill will. "The creature was trying to survive. It has that right, doesn't it?" he asks. But Kirk perceives more in Crater's interest:
You bleed too much, Crater. You're too pure and noble. Are you saving the last of its kind, or has this become Crater's private heaven, here on this planet? This thing becomes wife, lover, best friend, wise man, fool, idol, slave. Isn't a bad life—have everyone in the universe at your beck and call. And you win all the arguments.
It's not certain how accurately Kirk has Crater's measure here, but, even if Crater's personal motivations are complicated, the professor's points are valid. There is no simple right or wrong here, and the resolution, abrupt and violent, is fittingly hollow for all involved.