Why must the speed of light be the universal speed limit for all the fundamental forces of nature? Einstein discovered from studying the electrodynamics of moving bodies, that the speed of light $c_{em}$ is the same for observers, which move relative to each other with a constant velocity. But why must the speed of light be also the universal speed limit for all the other fundamental interactions (gravity, weak and strong force)? Why must $c_{em} = c_{g} = c_{weak} = c_{strong}$ ?
 A: Speed of light in vacuum is not a limit. It is fundamental property. Limit means light can move at any speed up to c. But light moves at exactly c, no slower, no faster. So, it is a fundamental property. 
Same way, the speed of forces (gravity, weak and strong force) is also a fundamental property in universe and has the same value as c.
The limit is only for matter because matter can move at any speed up to c but not >= c. 
c is also the speed (property) for energy (EM, gravitational waves) when it moves alone. But when energy moves matter (mass), then it moves at speed given by formula E = .5 * m * v * v. 
There are other scenarios for example water waves (or sound waves) transmit energy. The speed there is a property of the medium.
This way, the speed of light and forces can be considered a property of space.
A: $c$ is not first and foremost the speed of light. It is first and foremost the universal speed limit of a cause-effect relationship - if $A$ influences $B$ (in the same inertial frame) causally, and if $B$ is a distance $d$ from $A$, then the minimum time that must elapse before the influence can reach $B$ is $d/c$. Since the interactions you name are believed to mediate cause-effect relationships, $c$ limits their speed too. Another, evocative, often-used name for $c$ is "universal signalling speed limit".
The existence of such a $c$ can be inferred from basic symmetry and spatial isotropy assumptions about the universe. 
$c$ comes to mean the speed of light because it follows from special relativity together with this "more fundamental" notion of $c$ that all, and only, massless particles must always be observed to travel at the cause-effect speed limit, and that this observed speed is independent of an inertial observer's reference frame. So, given the "more fundamental" notion of $c$, the frame independence of the speed of light can be taken as an experimental confirmation that light is mediated by a massless particle. 
I say more about these ideas in this answer here
A: The main point of Einstein's relativity is that EM laws are the same for every inertial observer. Using this with the principle of relativity that tells that (inertial) motion is a relative concept you are able to prove that the speed of light is absolute. You can go further and show that transformations that relates two different inertial observer always preserve the interval:
$$
\Delta S^2=\Delta r^2 - c^2\Delta t^2
$$
where $\Delta r$ is the distance between two events and $\Delta t$ the time interval between this two events. You can think in this events as explosions that take place somewhere in space at some time.
Now, if you try to build fields that respect relativity, you always find a potential like:
$$
\phi (r)=\frac{e^{-r/r_0}}{r}
$$
You see that this $r_0$ provides a natural lenght when the interaction is important. If you are at scales $r\gt\gt r_0$, than you can neglect this interaction. This $r_0$ is related to the "mass" of the signals of the field, i.e. at related to the dispersion relation of the signals. The equation of this field in the vacuum is:
$$
\partial_t^2 \phi =-\frac{c^2}{r_0^2}\phi
+c^2\partial r^2 \phi $$
which tells that the velocity of a signal depend on the frequency. See here 
This means that fields with signals that travels with speeds different of light has a limitated range $r\sim r_0$ of importance. So, every interaction that survives through arbitrarily distances, $r_0\rightarrow \infty$, has signals that travel at the speed of light,  $v\rightarrow c$.
