Playing with a compass is good fun, I agree. You might consider what the comparable tool would be for a Linelander, or, for that matter, a Spacelander wanting to construct portions of spheres rather than portions of circles. The analogies are not perfect here--there is something special about circles.
Along these lines, there are some mathematicians who have carried out the exercise of imagining what geometry would be possible with other tools, such as and angle-trisector instead of an angle-bisector (which is one of the main functions of a compass. Can you think of others?) With such an angle-trisection tool, both of the problems you mention can be solved. It is also interesting to note that angles can be constructed if you allow a _marked_ straightedge and compass, if we can mark two points and slide the straightedge along so that one point is on a fixed line and another is on an arc of a circle centered on the line. I wish I could easily come up with a diagram to go here, but let's see if I can transcribe an argument that will enable you to reconstuct the needed diagram. Start with an angle B0A that we wish to trisect. First construct the line L through B parallel to 0A (so we are definitely in Euclidean geometry here). Draw the circle centered at B through 0. Put two marks on the straightedge indicating the distance from B to 0. Now, position the straightedge so that it goes through 0, and one of the marks is on the circle at a point C while the other is on the line L at a point T. Then the angle T0A is one-third the size of the angle B0A. You can show that angle T0A = angle TBC = half of angle BC0 = half of angle B0C, and that does it. Right? Let's see how to get the picture to make this clear. In any case, it shows that the restriction of a compass and an unmarked straightedge is a restrictive game indeed.
One of my students years ago, Judy Skorupski, made some beautiful models of regular polyhedra in transparent plastic and then sewed in the edges of the dual models with knots at the midpoints of faces. She then went on to a career as a high school teacher in Cranston RI. I wonder if she is still there. I think that her models are in a box in Dr. Reynolds' office. Ask her?
The kite is rather rigid, and in fact there are struts in the cubes that define cuboctahedra--do you see them in the picture? I think that you have to have something there to stabilize each of the cubes, but I know that you don't need that many struts. Exercise: How many small struts from a midpoint of one edge to another are necessary to make a cube rigid? How many would make a hypercube rigid, if that question makes sense?
I remember a collection of rings like that available in Chinatown in San Francisco many years ago. Does anyone have access to a Chinatown gift shop somewhere?
I'm not sure how you might want to respond to my responses--perhaps by adding another link after mine? If so, please mail me so that I know there is more to look at in your week 7 response.