If absorption and selective re-radiation of photons gives materials colours, shouldn't insulators be transparent? My question/confusion is with reference to the following answer, (I don't have enough reputation to comment on that specific answer)
https://physics.stackexchange.com/a/137311/150045
Shouldn't all insulators be transparent because their band gaps are too high to absorb any visible light?
As many insulators are coloured (e.g. wood), they should fall in the 4th case (coloured) mentioned in the answer, where 
"...excited absorber has a nonzero amplitude to transfer energy to the surrounding lattice..." 
(I assume this statement means that atoms absorb photons and transfer that energy to the lattice as vibrations, while atoms' vibrations, according to my limited understanding, is the movement of electrons between higher and lower shells/bands due to energy absorption)
However, using the mechanism in the question, they have too high of a bandgap (otherwise they would be semi-conductors or conductors?), so they shouldn't absorb anything and be transparent.
Is possible that photons are absorbed to change only the electrons' momentum within their same energy levels, thereby we only see certain colours (radiated back?), while others cause the change in momentum ? 
(Though this also seems unlikely, since then they would be able to conduct in valence bands?)
Also, where does white color fit in, according to the method detailed in the mentioned answer?
P.S, is it possible to bring this question to the attention of the poster of the answer I mentioned above?
Apologies for the long question. I just want to clarify what I understand (and where there could be errors) and where I am confused.
Thank you.
 A: I will tackle the title 

If absorption and selective re-radiation of photons gives materials colours, shouldn't insulators be transparent?

There is some confusion already. Absorption means the photon disappears by transfering all its energy to the lattice , as an example the lattice of a salt crystal,

This is a typical lattice, it has many degrees of freedom and energy bands, which are organized according to lattice coordinates, i.e. the little balls in the diagram oscillating in three dimensions around their rest point ( vibrating) or rotating around their base point. These give the vibrational and rotational levels of the lattice solid and will have quantum mechanical energy levels that can absorb photons and reradiate photons and are the main source of black body radiation. These energy levels are of lower value, infrared frequencies, then the energy levels of the bound molecules/atoms, which are of higher frequencies.
A photon hitting a lattice, if its energy does not match an energy level difference, will not be absorbed but will continue out with a little  deflection from elastic scattering with the fields it meets. These are the transparent materials.
The photon is lost in absorption, not reflected , its energy can go to vibrations of the atoms in the lattice, or distributed in exciting electrons of the atoms to higher energy levels. The relaxation  back to the ground level, may give off photons from the surface of the lattice of characteristic frequencies. Color is a matter of color perception and could have more than one frequency.
Reflection means that the fields on the surface are so strong that the photon scatters elastically backwards at an angle (think of a ball hitting a wall). If the surface is smooth, you get a mirror, if it is rough a shiny surface. If the photons can penetrate a bit and some be absorbed and some reflected , the color perception combination will change and the surface will appear colored.
In general for images to be retained either in transparency or in reflection there should be mainly elastic scattering. Absorption and re-radiation will lose the coherence of the images, because the re-radiated photon has a 4 pi phase space to scatter into.
Some lattices are transparent to some frequencies, i.e. any radiation cannot raise vibrational or eletron energy levels and can go through practically untouched by elastic scattering on any atoms on the way. These may be insulators, as glass, or conductors as here..

Shouldn't all insulators be transparent because their band gaps are too high to absorb any visible light?

The band gap is not a band gap for photons, but for electrons :

A photon might take an electron from an energy level to the band gap, and be absorbed, but how the electron will fall back to that energy level is a matter  of spherical symmetry, not reflection. It is a new photon that may or may not be reflected backwards depending on the material whether it is transparent or opaque. But this photon will not be the one transmitting an image. For transparency the photons should scatter little and only elastically, i.e. find no energy level, either in the atoms or  the lattice , to be absorbed in.
For insulators the band gap is large showing the difficulty for electrons to be excited to the conduction band, and is of the order of electron volts. 
Transparency is not directly connected with conductance or insulation. 
