How does a single photon behave propagating through glass? Take a normal pane of window glass and a photon somewhere within the visible range.
My limited understanding is that when light passes through glass one shouldn't think of little spheres travelling through the glass but must take into consideration quantum effects which somehow give a kind of statistical average resulting in the photons seeming to pass through the glass like little spheres.
What would happen if one photon propagated through the glass?
It wouldn't have the presence of all the other photons to reach some kind of average.
Direction of reflection and refraction can be predicted when large numbers of photons are involved.
I don't think the resulting direction could be predicted with one photon.
I imagine it would be absorbed by the first electron it meets and then re-emitted in a random direction.
Any better understanding on what would happen here?
 A: Each photon is describable as a wave until it is detected.  The wave is indeed scattered at every atom or electron in the glass, from whence it is scattered in all directions.  But the scattered portions of the wave all add together coherently and end up re-forming the original wave, over and over again until the wave exits the glass.  The square of the wave represents a probability distribution.  Downstream, a detector may detect the photon at a particular time and location.  But if the experiment is done over and over again, the photon's detection time and location will vary.  If you map all the detection locations and times, you will get a distribution of detection events that is the same as the intensity map of a light wave containing a large number of photons. 
This principle has been demonstrated many times in different ways, and is one of the experimental foundations of quantum mechanics.  Here is a fun link to a demonstration: single photon Young's experiment. It is a variation of the basic double-slit interferometer.
If you do a Google search for "propagation of a single photon through glass", you will find a lot of discussion of the topic.
A: A photon is a quantum mechanical point particle in the table of elementary particles, interacting primarily electromagnetically.
A beam of light passing through the glass is a superposition of zillions of photons. The fact that images and colors are retained means that the individual photons can only be scattering elastically with the whole "lattice" of the glass.(lattice in quotes because glass is not organized as well as crystals)
I.e. the photon interacts with the whole glass it passes through, as an elastic scatter that retains the phases with the other zillion photons so that the images can be transmitted with minimum distortion. It is not absorption and reemission, it is a specific quantum mechanical wave function that allows this .
The double slit single photon experiment may give an intuition:

The scattering is "single photon scatters off two slits given width given distance apart". On the left the individual photons seem random. On the right the accumulation builds up the classical light interference pattern from two slits, i.e. the probability distribution of the photons gives the classical interference pattern.
A: Are you familiar with the concept in quantum mechanics of particles being in a superposition of many states at once? When a single photon travels through glass it is in such a superposition. The single photon comprises an entire beam of light, which interacts with many atoms simultaneously as it propagates (according to classical electromagnetic theory). Only if there is a "measurement" (i.e. the photon is absorbed into a state which kills the coherence) does the "wavefunction collapse" onto a single atom. The photon being absorbed into a single atom buried in the glass would then provide a constraint on the path the photon took to get there. A photon cannot be thought as a ball zig-zagging around between glass atoms, for then it could take an arbitrarily long time to arrive at the destination. Rather, it must travel at the speed of light (divided by refractive index), point-to-point to minimize travel time. No; it's better not to think of photons as balls at all. All there is is the electromagnetic field. A photon is simply the manifestation of a single quantum of energy, momentum, and spin being transmitted from one place to another. 
