Is it possible to design a 2D periodic structure so that it reflects a specific wavelength? Is it possible to design a 2D periodic structure so that it reflects a single and specific wavelength at normal incidence? Is it possible to do such with dielectric structures so that the overall structure has a perceived 'color' when seen in reflection?
 A: You know, I believe this is possible (for some flexible interpretation of a “single specific wavelength”). Check out dielectric metasurfaces. It is a recent topic of interest among the optics/nanophotonics community, which relies on 2D planes filled with nanoscale dielectric resonators. These structures can be engineered to impart an arbitrary phase to the light at any point in the plane for a given wavelength, which is useful for creating flat lenses, polarizers, pulse shapers, and other passive optical elements. So reflecting a single color should be within the possibilities, although the reflected beam may not end up normal to the surface, even if the incident beam is.
And then, of course, there are Bragg mirrors, which are composed of a stack of thin films, and which might be considered 2D if “2D” can mean micrometers thick. These can be engineered to reflect certain wavelengths and not others.
A: I don't think this is possible, if the structure is an idealized dielectic confined to a two-dimensional plane perpendicular to incidence. If the time variation of the incident wave is $f(t)$, then the time variation of the reflected wave is just $\alpha f(t)$, where $\alpha$ is a constant. To get a frequency dependence of $\alpha$, you need things like resonances in the material, and we do see such resonances, e.g., in glass, but they don't act like notch filters.
You can make a sinusoidal reflection grating, but that doesn't select a particular wavelength. It will just produce $m=-1$, $0$ and $+1$ fringes, and no other fringes. The transverse variation in the surface doesn't produce filtering.
A: A Bragg grating is highly wavelength selective.  We routinely make Bragg gratings whole wavelength selectivity is just a few nanometers.  Note that the selected wavelength is angle dependent: a different angle of incidence corresponds to a different selected wavelength. A Bragg grating needs to be at least some tens of microns thick, though to achieve this degree of selectivity.  That probably doesn't qualify as a 2D structure.
One way to make a highly wavelength selective, reflective, nearly 2D structure for a beam having normal incidence, would be to couple incident light into a waveguide mode where it could interact with a 2D grating, be diffracted within the waveguide, and then be coupled out into a direction opposite to the incoming beam.  In essence, this would use a 2D Bragg grating in the waveguide. There might be some other approaches, but this one can be done relatively easily.
