About the lowering of the speed of light in non-vacua I don't understand really why matter interacts with light so as to slow down wavefronts to a speed strictly below $c$, but still preserving that sharp wavefront-like signal.
This is somewhat unexpected if you assume that such behaviour must arise from a field $u$ which, at the microscopic level, must obey a hyperbolic local evolution equation (a wave equation with damping terms) of the form $(\partial_t^2-c^2\Delta )u+\text{lower order damping terms}=0$: you would expect the wavefront to still travel at speed $c$ while getting attenuated exponentially in time (*). It is very difficult to conceive of a dynamics so fine-tuned that in the wake of that advanced wave-front there would grow another sharp wavefront (which would be supposedly the one observed by the experimenter).
Is there an account or an argument why experimenters nontheless report on a speed of light in such media? Is it so that light wavefronts remain relatively sharp and why?
Let me indulge you with a vague attempt/idea of my own: Conceive of the position $x(t)$ of individual light particles as random variables which after some time $t$ has passed is well-described by a sum of i.i.d. increments $\Delta x_j$ (the number of such terms in the sum is proportional to $t$ and the physical interpretation of this sum is that it should arise from the quasi-independent "scattering-or-evasion" events that the particle encounters along the way). According to the law of large numbers, the mean position of many light particles grows like $t$ while according to the central limit theorem the standard deviation of those positions grows only like $t^{1/2}$. So a wavefront made up of several particles may preserve a degree of sharpness as far as the ratio of its width divided by the distance to the emitting light source is concerned.
(*) Thanks and shout out to Jhor for raising my awareness of the notion and experimental verification of so-called pre-cursor waves. Also see here.
 A: Too long for a comment.
In contrast to @Annav answer, i strongly assert that there is absolutely no quantum effect in this problem (by the way,  the rigorous theory of QED in dielectric media has still some problems).
I think that the OP has all the key ideas for the problems of propagation speed, except that it  is barely related with damping or with spreading by dispersion.
As the question mixes the different aspects, it is difficult to answer accurately.
Here I address the question of speed. In fact, in real experiment, and for sharp enough pulses one can see a small fraction of the wave which propagate a vacuum velocity $c$, another small fraction which propagate at the phase velocity $v_\phi=c/n=\omega/k$, while the main pulse propagates at the group velocitie $v_g=d\omega/dk$.
(Notice that in standard media $0< v_g< v_\phi< c$, but the justification of theses inequalities is another question).
This phenomenon is known for a very long time as "Sommerfeld precursors". You could find a lot of internet links about them, and a detailed analysis in old editions of Jackson book.
If your question is more focused on spreading, please ask another question with a title like spreading of width if pulse or so.
Edit generally speaking, disipersion Is NOT a diffusive effect, and therefore do not follow the $t^{1/2}$ suggested in the OP.
And if you want to know why the speed is smaller than $c$, do the same.
A: From my experience with ultrafast femtosecond lasers and fiber optic, you are right assuming that the sharp wavefront gets distorted progressively as it travel thru transparent material.
Light slow down in transparent material for similar reason electric signal in cable slow down. Electric engineers use as model of cable a large (infinite) number of inductors in series and capacitors in parallel.
An impedance matched cable will attenuate a signal but avoid reflexions. Each pair of inductor and capacitor can be viewed as a band pass filter. The phase of the wave is changed with frequency. Each part of the cable absorb the energy then release it after a given time.
Electromagnetic waves in transparent material encounter an LC like system that capture the energy and release it after a short delay, making the crest of the wave trailing, which over time delay the signal while preserving frequency.
This is a spring/mass delay system where the mass is the electron and the spring is the electrostatic force from chemical bond.
Here is in detail how the electromagnetic wave change speed... Instantaneously:
The electromagnetic wave slow down right at the interface between vacuum or air and the transparent material. The wave don't decelerate from 100% of c to, for example, 66% of c. It adopt the new speed immediately. Because mass is zero, acceleration tend toward infinity.
Suppose we zoom on a single cycle of a wave, let's use the 360 degrees notation. As the wave start entering the first layer of atom/molecule in the transparent material, let say the first 45 degrees, that portion already travel at 66% c while the rest of the wave from 45 degrees to 360 degrees, still outside, continue at full speed, 100% c.
Some femtosecond later, half the wave reached inside the material traveling at 66 % c and the rest, from 180 to 360 degrees keep coming at 100 % c.
Eventually, over the time of one wavelength, the electromagnetic wave is fully inside the transparent material and will keep travelling at 66% c until reaching an exit side.
Note that, contrary to objects in viscous liquid which feel a constant drag and lose speed over distance, light doesn't change speed, but keep a constant speed even in a 40 km long fiber optic.
When exiting the transparent material, each part of the wave immediately change speed from 66% c to 100 % c, assuming air or vacuum. There is no acceleration but only an instantaneous change of speed.
A: This should be a comment, 
In mainstream physics, particles of light are called photons, and are axiomatically defined in the standard model of particle physics.
Classical electromagnetic waves are a superposition of a huge number of photons, all travelling with velocity c because they are zero mass. In a transparent medium, which is the one you are discussing, the individual photons travel longer distances at angles with the optical ray, but still retain their phases and superposition of the wave function. Their longer path length on average is responsible for the lower group velocity of light in the medium.
The light fronts remain because in transparent media the photons interact with the whole lattice, and the phases are retained. See here how in QED classical fields emerge from the quantum ones.
A: In the context of qed, photons are slowed down in the medium because they are absorbed and re-emitted by electrons. May I be forgiven for repeating what I recently wrote in an answer on reflection by a mirror? (The reason is the same although the situation is different.
Although a single photon can only be absorbed and emitted by a single electron, it leaves that electron in exactly its original state. There is no record, and no way of knowing, which electron absorbed and emitted the photon. According to quantum theory, to calculate the result when any electron could have absorbed and emitted the photon, we must form a superposition of all the processes which could have taken place. The calculation in quantum mechanics takes the form of wave mechanics -- this does not have to mean that there is actually a wave, only that the mathematical theory behaves as though there was a wave. The consequence to the wave its speed of propagation is slowed down, resulting in the observed refraction just as for the classical argument.
