I think that "shuttlecraft" is also the word when describing them in plural - I believe there is a reference in "The Omega Glory" to the effect that "all four shuttlecraft" were still aboard the Exeter.
Yup, like deer being singluar and plural. "Shuttlecrafts" sounds awkward.
"The shuttlecraft is inoperative."
"So take another."
"The shuttlecraft are inoperative."
"All of them?"
Set changes are never something I use to "prove" we're seeing multiple rooms. Otherwise we'd have multiple engine rooms or sick bays. Obviously the walls were changed for production and script reasons. Trek was reasonably consistent, but watch Lost in Space or Voyage to the Bottom of the Sea to see rooms, walls and control panels changing weekly. It's not like there were multiple Jupiters 2. You have to draw the line somewhere between internal reality and "needs of the story."
One transporter room in the original series. Otherwise, they could have beamed down both landing parties simultaneously in The Apple. "Meet me in THE transporter room." If there were more than one, someone would have asked "um, which one, Cap'n."