Rainbows at other wavelengths

Fascinating question. But if you mean rainbows in other wavelengths from the Sun, yes to UV and IR, but no to Xrays. Simply because there's no significant radiation from the Sun at Xray wavelengths (see this graph for radiation per wavelength).

Most of the Sun's radiation is at visible wavelengths, which I guess is no coincidence. Life on Earth has evolved to use the light available!

There are definately detectable rainbows at other wavelengths tho. See here for pics in UV (slightly inside the visible rainbow) and IR (slightly outside).

Yes. It is possible to produce rainbows of Ultraviolet or X-Rays in the lab. The SLAC team has already built a tool to produce rainbows of X-ray.