When Einstein introduced special relativity (SR) in 1905, there was no idea of a "photon." In fact, he defined a stationary frame as one where "both Newton's mechanics and Maxwell's equations" applies. However, Newton's Laws are not a precise description of the universe even in the highly modified relativistic version derived in 1905. There is another theory that has replaced it called quantum mechanics. Quantum mechanics has Newton's mechanics as an approximation certain conditions.
In a book that Einstein wrote on relativity in 1922, he defines an inertial frame as that where "the laws of physics are simplest." Quantum mechanics was just being developed. So I would say that this "new" version of SR has the assumption that an inertial frame is defined as one where Maxwell's equations and a relativistic version of quantum mechanics apply.
One of the major predictions of quantum theory is that a pure wave can collapse into a pure particle, and vica versa, because of certain processes called "measurements." However, a pure wave and a pure particle can not exist at the same time. Einstein himself made up the concept of photon, which is the particle that a wave of light collapses into when one tries to measure "where" the energy is. I went to the concept of photon because you may be more familiar with the idea of "particle" having a mass than a "wave" having a mass.
One amazing thing in 1905 was that Einstein was claiming that the wave had a type of mass. One could work out an "energy density" of a light wave proportional to intensity. However, that is difficult for some people to conceive.

Well, sort of both.
If you insist on thinking of photons, probably the best representation is as a spinning ball moving at speed "c" in some direction. The axis of spin is in the direction of motion. It can go either clockwise or anticlockwise. There is no wave in this representation.
Lets look at waves. Instead of imagining a wave made of photons, imagine a wave made of electric field lines that point in a certain direction. The "strength" of the electric field is really similiar to length. At a particular point, an electric field can grow, then shrink, than grow in the opposite direction, than shrink, etc. This is called plane polarization. Another possibility is that the electric field can remain the same length and simply move around in a circle. This is called circular polarization. There is no particle in this prepresentation.
The two directions of spin of a photon are mathematically related to the two types of circular polarization. However, the relationship isn't obvious without knowing the theory REALLY well.


According to quantum mecahnics, you can not measure both angle and angular momentum at the same time. This is because of the following. The spin has to do with wave measurements and the angle has to do with particle measurements. Since we are assumming that we know the angular momentum of the electron exactly, the electron is acting more like a wave. It can't coexist with a system that will be measured as a particle, which would be the angle of something on the electron. Therefore, since the angle can't be measured, one can not measure the "speed" of spin either. It can be slower than light, or faster than light, but who would know?

Some people did a calculation using questionable assumptions that the "edge" of the electron was going twice the speed of light. However, it can't be measured so SR has nothing to say about it.