Is the Speed of Light an universal spacetime constant, the velocity of electromagnetic waves, or of photons? This question has been touched tangentially by What's a better phrase than "speed of light" for the universal spacetime speed constant? and Could light travel more slowly than the "universal speed limit"? Could this imply quantization of spacetime? but I do not think they reach the core of my question.
When I read that many other theories, e.g. the Standard Model, have been constructed to be Lorentz invariant it appears that electromagnetism has some of preponderant role in the universe compared to other theories.
I suspect that the answer is that this is not the case, and that the space-time Speed of Light (with upper-case) is really more of a universal constant that is not necessarily related to the speed of the electromagnetic waves or of photons.
There are many other related questions: 


*

*would it really be devastating for relativity that we find out in the future that photons have a tiny mass and move slower than ... uhh... the speed of light?

*which "c" is applicable in different situations? I mean, possibly the space-time constant should be used in $E=mc^2\gamma$, it would be the  velocity of gravitational waves, etc..., but in other situations such as the Bohr radius or Compton wavelength I am not sure.


Well, any answer will be appreciated. Please tell me if you know any reference to bibliography that address this issue.
 A: 
it appears that electromagnetism has some of preponderant role in the universe compared to other theories

This is not true. It played an important historic role, but is in no way theoretically "unique" because the photon travels at speed c. Indeed, the gluon also travels at speed c. If photons were found to be slightly massive, it would change a lot of things, but it wouldn't change relativity. At this point we have a well constructed theory where all and only massless particles travel at speed c. In other words, the photon is only historically central to relativity, not conceptually.

the space-time Speed of Light (with upper-case) is really more of a universal constant that is not necessarily related to the speed of the electromagnetic waves or of photons.

You've pretty much answered your own question, because this is exactly right. Historically, questions regarding inertial reference frames and electromagnetism led to the development of special relativity. Within relativity, an important speed constant appears. If indeed the photon is massless, then this speed should be the same as the speed of light. Even if the photon were to turn out to be massive, this in no way obligates us to rewrite our theory. It just means that the historical tool we used to build relativity wasn't exactly what we thought it was, but was close enough for the early development of the theory. Our current theoretical understanding of Minkowski space (the space-time geometry described by special relativity) is not one that depends on electromagnetism, in fact it's the other way around : in our current understanding of electromagnetism, we use our understanding of Minkowski space to conclude that massless particles like photons should move at the speed of light.
A: A quick alternate perspective:


*

*Not really. Relativity only requires the existence of an invariant speed $c$, it doesn't require that anything actually travels at that speed. So if photons were massive, there would be no problem, although some results in cosmology might have to be modified a bit.

*Pretty much every time you see it, $c$ means the invariant speed. So if photons were discovered to be massive, we would have to use some other symbol for the speed at which light actually travels - and note that it wouldn't be a fixed constant anymore, because in that case the speed of light would be reference frame-dependent. In some circumstances you'll see $c$ used to denote the speed of a wave, which may not necessarily be the invariant speed of relativity, but those are relatively rare and usually easy enough to pick out from context.

A: 
1. would it really be devastating for relativity that we find out in the future that photons have a tiny mass and move slower than ... uhh... the speed of light?

As the phrase "... uhh ... " in your question anticipates:
there is some devastation lurking in that question; namely a contradiction to the essential understanding of "light" as "any signal whatsoever, associated with the signal front" for the purpose of defining geometric (or kinematic) notions such as  


*

*mutual rest (a.k.a. as joint membership in an inertial frame) of participants,

*duration, or at least: duration ratios,

*distance (between participants who are at rest with respect to each other), or at least: distance ratios, and

*speed;
not even to get into subsequently defined notions of dynamics, such as "mass", or "refractive index".
The close association of "light", in the sense explained above, with the exchange of signals between electro-magnetic charges is just due to (systems constituted of) electro-magnetic charges being especially common and conspicuous in our particular corner/era of the universe.

2. which "c" is applicable in different situations? [...]

The "c" which is introduced foremost as a (non-zero) symbolic coefficient in the definition of "distance" as "c/2 ping duration" always appears as signal front speed; consistent with the meaning of "ping" and the corresponding duration.
In Einstein's initial approach to relativistic geometry and kinematics the definition of (how to measure) distance still appears inverted (and not coordinate-free):
"In accordance with experience we shall assume that the magnitude
$\frac{2\ \overline{AB}}{t'_{A}-t_{A}}=c$, where $c$ is a universal constant [the speed of light in vacuum]."
The understanding of (how to measure) "distance" as a notion which has to be defined in the first place has been expressed subsequently by Einstein arguably in the prescription that 
"All our well-substantiated space-time propositions amount to the determination of space-time coincidences [such as] encounters between two or more recognizable material points"; 
and most explicitly perhaps by J. L. Synge ["Relativity. The general theory", p. 108]:
"For us time [duration] is the only basic measure. Length [distance] is strictly a derived concept". 
A: all three move at the same constant speed that is c. the reason to prove it is that gravity waves travel at the same speed. And even the effects of gravity travel at speed c. That is the reason why I say that spacetime's structure can only change at this speed. Am I right to say that the speed of light is finite because spacetime's structure itself needs to somehow accommodate EM waves passing through it? Am I correct that nobody has yet combined gravity with QM? In QM currently they say every particle has its own wavefunction that describes its probability distribution at and around its coordinates in space. GR explains the effects of gravity, but it does not explain how they work. Since the EM waves can only propagate through space with speed c, and the effects of gravity can only propagate with the same speed, and even gravitational waves can only propagate with the same speed, it would be accurate to say that information can only propagate through space with speed c.
Now the reason for that is that space itself can only change its own structure at this speed. Why? Lets combine GR with QM. Lets say not only particles have their own wave function, but space itself also "stores" a wavefunction for each of its spatial "coordinates" or position. So in space each 3D coordinate would not only be a spot, but it would also have a wavefunction itself. Just like how QM also says space is not empty, it has Q Fields everywhere in it. So if it has Qfields everywhere in it, it can have a wavefunction everywhere.
This wave function will do the same for a certain spatial coordinate, as the normal wavefunction(that belongs to the particle itself) does with the particle, it just shows the probability distribution of the particle at its own coordinates, and around it in space. So space itself would not only have Qfields at every single spatial coordinate, but also a wavefunction. Lets say there is a wavefunction for point A and point B. If point A and B have no particles at their positions (so they are "empty") then the wavefunction for point A will show a neutral distrubution of possibilities (0 most likely for its own coordinates and the surrounding spacecoordinates) .
The wavefunction for point B will show the same. Now if a photon enters point A, the wavefunction of point A will start changing and will first show higher probabilities at its surrounding spatial coordinates in the direction the photon is coming from, and then as the photon passes through point A, the wavefuction of point A will show higher probabilities along the way the photon passes through it and its spatial surroundings, eventually at the center coordinates of A and then in the direction of the photon's passage.
Eventually as the photon passed through point A, the wavefunction for point A will show lower probabilities at the spacial surrounding coordinates from where the photon came, and at point A's center coordinates, and will show higher probabilities in the direction the photon passed on. Then, as the photon moves towards point B, the wavefunction for point B and its spatial sorroundings will start showing higher probabilities in the direction the photon is coming from (that is the direction from point A), and as the photon passes through point B, it will show higher probabilities along the way the photon passes through it. Eventually the photon will pass through point B and will continue it's way towards a point C. At some point, the wavefuntion of point A will become neutral again, showing 0 probabilities for its own coordinates and all its spatial surroundings.
Now let's combine this with gravity and GR. Lets say there is a huge gravitational mass at point B. Point A is "empty", but still inside the huge mass's gravitational field. So the huge mass at point B will have a gravitational effect on point A. How? well, the wavefunction at point B will not be neutral, since the effect of the huge mass at point B will have an effect on point A too! It will change the wavefunction at point A too! What will the wavefunction at point A look like? It will show higher probabilities in the direction of point B, where the huge mass is! And lower probabilities for its own coordinates, and in the other direction (not towards B). So how does gravity work then if we combine GR and QM? Lets say a photon is arriving from point C, which is even further from the gravitational mass.
As the photon makes its way to point A, it should just pass through as in the example before, but it will not! Why? Because in the example before, A and B were both outside any gravitational fields, and had neutral possibility distribution values for their own coordinates and their spatial surroundings too. But now, there is a gravitational mass at point B, that affects point A's wavefunction too! So as the photon wants to pass through point A, it enters into the wavefunction of point A, which shows higher probabilities towards the direction of point B, the huge mass.
So as it passes through point A, its own wavefunction (3D or 4D matrix of ditribution of possibilities for point A's center coordinate and its spatial sorroundings) will combine with the photon's own wavefunction (thats already in QM, just another matrix of ditribution of possibilities for the photon's actual coordinate and its spatial sorroundings). So as the two matrices combine, the wavefunction for point A will change the direction of the photon as it passes through, just like GR predicts. And just like QM predicts, it will all be described by the combination of wavefunctions, and distribution of probabilities. The QM's current wavefunction will also stay in tact since the photon will keep its own wavefunction too.(that should look like a momentum vector, showing higher probabilities towards its target direction) Is that a possible solution?
A: From a modern point of view, it is not neccesarily any of those things. Since relativity treats space and time on equal footing, the constant c serves the purpose of converting seconds into meters. In that sense, it could be considered a universal constant because we measure spatial distances with different instruments than temporal distances.
The connection of c to the speed of light is mainly historical. Einstein wanted to explain maxwell's equations and did so by abandoning Galilean invariance.  Maxwell's equations are only exactly correct (and light only propagates at c) if the photon is massless. 
It's possible to create a relativistically correct theory of electromagnetism in which the photon is not massless and therefore photons are like every other massive particle and Maxwell's equations are only approximately correct. This was first done around 1913 and is known as the Proca Lagrangian.  The experimental limits on the photon mass require the mass to extremely tiny if it is not massless. Right off hand I don't know the most recent limits.
