Welcome to our group, Bob! The type of project that you want to do can be accomplished using a number of cutters. 3-Ply Bristol board is most likely the most challenging media you would be cutting for this project. While the Cricut Maker can cut 3-Ply board with the special knife blade, it does take a while. I have never seen this type of board available in a self-adhesive format. You could apply xyron or double sided sticky tape to the back of the pieces after cutting. I have been able to cut up to 4-ply board using the deep cut blade on the Scan N Cut and on the Cricut Maker with the craft knife. Keep the speed slow. Multiple passes will be necessary. Cricut may take a lot longer, as it requires many more passes. Cricut Maker's are not perfectly precise, but likely will be close enough for your project, within 1mm. Slower cutting with multiple passes will keep your cuts more accurate, whichever machine you use.
Thin leather or bookbinder cloth should cut well on any of the cutters.
I have cut very thin plastic up to .07 on a number of my machines with no problem. Thicker plastic and acrylic do not cut so easily with the machines.
Your project sounds very interesting, especially since I am a musician. We look forward to seeing photos of your completed project!