That's a good question about the DVD plastic fluorescing. I don't know of anyone who has raised that question about the Public Lab spectrometers. If you have access to those pure UV light sources, you could answer the question. Shine only the pure UV source on a DVD and take a photo with a good camera. Good cameras have UV block filters, and the glass in the lens will block most UV, so if the photo is black then the DVD does not fluoresce in that UV light.
If the slit fluoresces in UV light, just make a slit with two razor blades or utility blades. That will work better then the printed slits.
The web camera in the Desktop Spectrometry kit won't resolve the peaks as well as the Ocean Optics spectrometer, but if you use a good camera to take photos of the diffraction patterns your wavelength resolution will be much better.
Comparing the intensity of different peaks in a spectrum is fraught with error when using a DIY spectrometer based on a consumer camera, so identifying the minerals responsible for the peaks might require some guess work.