I cannot think of any reason why a material's absorption/emission through the visible spectrum would determine its absorption/emission through the infra-red spectrum.
I dunno, how hard is it to get materials that reflect radio waves?
It's probably reasonable to talk about materials that reflect almost all the light in frequency bands of non-negligible power as effectively being actual white.
Yes, you're describing a white body, the opposite of an ideal black body. It does not exist, of course. But we can try to approximate one with materials, at least for IR through UV. Now, if we do have a pigment that can reflect X-rays... I'd definitely want my roof painted with that.