Conceptually, I understand what light is, but I don't know what a photon would "look like" if it could be frozen in space/time. For instance, the notion of amplitude seems to be absent when discussing light, even tho it is drawn as orthogonal sine waves. Is the sine wave just a way to represent the periodic change in field strength, or do the fields occupy a volume such as would be generated by rotating a sine wave about its axis? Does a light quantum have length, or is it only an instantaneous value at a point in space (and then how can it be red- or blue-shifted)? How does the magnetic field component satisfy the requirement that all field lines be closed? Does light even behave like this on a discrete level?
One has to distinguish the two frameworks: the classical, light; the quantum, photons.
The classical electromagnetic wave, of which visible light is a part of the frequency spectrum, emerges out of zillions of photons, the quantum of light. This happens because the photon has an energy E=h*nu, where h is the Planck constant and nu the frequency of the classical wave that will emerge from zillions of photons . It also has a spin and can be described by a quantum mechanical wave function that allows the build up of the classical wave from the quanta. The energy of the classical wave is the addition of the individual photon energies building up an amplitude.
Is the sine wave just a way to represent the periodic change in field strength, or do the fields occupy a volume such as would be generated by rotating a sine wave about its axis?
The sine wave and cosine wave belong to the classical electromagnetic wave. Not to individual photons. Individual photons have a probability of appearing in space that is described by a sine/cosine wave, but not a distribution of its energy in space time. The photon's energy is one whole quantum. The emergent classical wave has a sinusoidal energy distribution in space time of the same frequency .
Does a light quantum have length, or is it only an instantaneous value at a point in space (and then how can it be red- or blue-shifted)?
The photon has no length, it is an elementary particle . The shift in frequency that is assigned a change in color is an emergent effect from the zillion of photons. For the individual photon, a red shift means a lower frequency/energy h*nu, a blue shift a higher frequency/energy h*nu.
How does the magnetic field component satisfy the requirement that all field lines be closed?
The photon does not have a magnetic field component, it is characterized by the potential entering Maxwell's equations and it will build up the magnetic and electric field components of the emergent classical wave. By itself it is a particle characterized by a probability distribution for its space time location.
Does light even behave like this on a discrete level?
On a discrete level, in the double slit single photon experiments which show interference patterns by individual photons, i.e. the probability distribution manifesting in the build up of the experiment, the emergence of the classical wave that coincides with the quantum frame is displayed clearly.
The only person I know, who spoke about the structure of the photon, is de-Broglie, listen to him:
For both matter and radiations, light in particular, it is necessary to introduce the corpuscle concept and the wave concept at the same time. In other words the existence of corpuscles accompanied by waves has to be assumed in all cases. However, since corpuscles and waves cannot be independent because, according to Bohr’s expression, they constitute two complementary forces of reality, it must be possible to establish a certain parallelism between the motion of a corpuscle and the propagation of the associated wave.$_1$
Many of them don't agree with him, thinking the quote is of 1929. Let me remind them that, we are still agreeing de-Broglie's hypothesis of matter waves as I know, which is actually derived from the above concept. So, the above quote is still valid. To know more from him, read the noble lecture from the below link.
Another legendary is Feynman, listen to him:
...Some other important observation are that, as light goes from one place to another, it goes in straight lines, if there is nothing in the way, and that rays don't seem to interfere with one another. That is , light is crisscrossing in all directions in the room, but the light that is passing across our line of vision does not affect the light that comes to us from some object. This was once a most powerful argument against the corpuscular theory; it was used by Huygens. If light were like a lot of arrows shooting along, how could other arrows through them so easily? Such philosophical arguments are not of much weight. *One could always say that light is made up of arrows which go through each other!*$_2$
The last line gives a brief description of how light behaves physically.
The below statement is again from the same specialist:
Light is something like raindrops-each little lump of light is called photon-and if the light is all one color, all the "rain drops" are the same size.$_3$
We see the world because of light, then what do you use to see the light? Can we use electron waves? That's upto you now to discover.