I work as a consultant to the NYC School Construction authority. We usually inspect low-voltage applications for temperature controls and the associated equipment. Typically, whenever we see a thermostat in the wall, it has a j-box and conduit running from the box to the drop ceiling, and the wire is then in midair (with anchors) till it gets to the vent motor. I have also seen instances where there is no sort of conduit, and the wire is run behind the sheetrock unprotected.
Is there anything in the NEC or somewhere in the NYC Building/Electrical codes that require any sort of cable to be run fully through conduit through a drop ceiling or behind sheetrock?

May I suggest that you call the electrical inspectors that are responsible for this school.

Think of it this way. No one is going to get hurt jabbing a nail into a thermostat line. It's low voltage power limited and usually no one cares whether it's protected or not

That conduit to above the drop does do a few things. 1) install is more rigid and professional, 2) New wires can be run easily, 3) It's protected.

I usually put a cord connector above the ceiling too. This would also serve as a fire stop.

Under NO Conditions should wires be laid across the suspended ceiling. They should be suspended in hooks from beam clamps and rods or in a wire raceway.

If the ceiling is a return air space then plenum cable should be run.

The NYC Construction Authority needs to have guidleines for construcion workers to adhere too and that needs to be well known.