The girl playing the wheel is Deborah Downey. Along with Charles Napier, she sang her own part. The performance scenes were lip-sync'd to a playback tape, but that's really her singing on it. She went on to become a painter, and one of her commissions was a portrait of Gene Roddenberry.
"Girl #2" was played by Phyllis Douglas, a favorite ST hottie of mine. She was on the show twice and got to ride the shuttlecraft both times. She probably thought they used it in every episode.
Nice trivia, and I agree with you about the lovely Ms. Douglas. Sadly not much of a part, and not much of an episode.
I once met Mary Linda Rapelye (Irina), who would have to be one of the friendliest, warmest people you could imagine. Still looked absolutely gorgeous decades later. If I were Chekov, I would have mutinied