Does non visible light such as x-rays or ultraviolet experience refraction and is their refraction index affected by their wavelengths (in their own spectrum)? Do they experience dispersion like visible light?
Yes. UV is beyond the blue and violet after a prism. But glass is not transparent very far into the the UV, the rest gets absorbed. For vacuum-UV (beyond about 10 eV photon energy) there are no materials to make prisms of.
At much higher photon energies, in the x-ray region, materials become more transparent again. But the refractive index is very close to 1. And actually it is a tiny bit smaller than 1 (away from absorption edges). This is used to make mirrors for grazing incidence that have total reflection.