Making models of the three-dimensional polyhedra is certainly a good start. Perhaps we should get everyone in the class to make a dozen or so standard-sized tetrahedra or octahedra or dodecahedra and then begin to assemble the fold-out versions of the regular polytopes in four dimensions? What about three inches on a side? Is everyone game for that exercise? Of course we still have to worry about the ways of sticking them together but I suppose we can find appropriate ways.
With respect to the 600-cell, which I agree is a tour de force, I think I see how people might have got to it. You know there is a possibility of doing the construction as soon as you realize that you can fit five tetrahedra around an edge with a little room to spare. So you can start with a tetrahedron and try to build outward in some systematic way, hopefully with enough of information to help you know when you are done! Perhaps it's better to start this exercise by constructing the 16-cell, with four tetrahedra around each edge--that seems more accessible. As it happens, it is better to construct the 600-cell by using long snakes of tetrahedra wound around each other. Oddly enough, it is somewhat easier to construct the 120-cell if we can come up with 120 dodecahedra of the correct size.
It would be nice to have a 3d spirograph, I agree, but most of the drawing implements I have seen are woefully 2d. As it happens, my sister-in-law gave me a "gravity graph" for Christmas, a somewhat modern version of a sand pail on a pendulum, for drawing Lissajous figures, the sorts of things that show up all the time on oscilloscopes. They aren't the same as your trochoids, of course, but the principle is similar--a kinematic way of generating curves by the trace of a point in motion.
I had success with one of the three websites you indicated, namely the Vanderbilt trochoid machine. The last one was busy and the first required some installation that I found unhelpfully documented. Did you get them to work?
Yes, the hexaflexagon is something that can be considered, in itself and in its possible higher dimensional analogues. Do you have one to bring in?