Do electromagnetic fields have inertia? Or, what sets the speed of light? In all mechanical waves, there is a restoring force and an inertial influence. For example, a plucked string oscillates because the restoring force brings it back to being straight and then the string's inertia carries it on past being straight and on to the other side. The mechanical wave speed is
$$v=\sqrt{\frac{restoring}{inertia}}$$
Are electromagnetic waves in some sense analogous? One view of electromagnetic waves is that the electric and magnetic fields each act as restoring forces for the other. For example, in Do the electric and magnetic components of an electromagnetic wave really generate each other?, Andrea wrote that "... the fields try to mutually reduce each other rather than generate. In fact they do that so well, that there is an overshoot, and the cycle repeats." This reply suggests that the electric and magnetic fields each act as restoring forces on the other one. But then, what limits the rate of this restoring and also causes the overshoot, hence acting as some sort of inertia in the system?
Another way of asking this same question is what limits the speed of light? What prevents electromagnetic influences from propagating a lot faster than they actually do? I can do the math that shows that the speed of light is
$$c=\sqrt{\frac{1}{\mu_0\epsilon_0}}$$
but I don't understand what's actually meant by the electrical permittivity or magnetic permeability of space well enough to get a physical understanding of how they set the speed of light.
 A: Here is an approximate analogy which may be useful, and I request the experts here to add their perspectives. 
An empty vacuum exhibits the property of supporting the propagation of electric fields through it. This means when we build ourselves a capacitor out of two conductive plates and assert a current through it in a vacuum, it exhibits capacitance even with no dielectric disposed between the plates. That capacitance opposes changes in the voltage across the plates. 
Similarly, an empty vacuum exhibits the property of supporting the propagation of magnetic fields through it. This means when we build ourselves a coil and assert a current through it in a vacuum, it exhibits inductance even with no magnetic material in its core. That inductance opposes changes in the current flowing through it. 
Using a mechanical system as an analogue, a capacitor becomes a spring with compliance and an inductor becomes a mass with inertia. This means we can think of the "springiness" of the vacuum as having something to do with its capacity to propagate electric fields and the "massiness" of the vacuum as having something to do with its capacity to propagate magnetic fields. 
The characteristic of the vacuum having to do with its ability to propagate electric fields is called the permittivity of the vacuum. The similar characteristic for magnetic fields is called the permeability of the vacuum. So the springiness of space comes from the permittivity and the massiness comes from the permeability. 
This means that like a piece of rope (which has a certain amount of springiness per unit length and a certain amount of mass per unit length) upon which we can assert traveling waves, free space has a characteristic impedance which is equal to sqrt(permeability/permittivity) and a wave propagation speed which is equal to 
sqrt (1/permeability * permittivity). J.G.'s answer below furnishes another interesting way to arrive at the same conclusion. 
We have no a priori reason to expect that the measurements of the vacuum permeability or permittivity should be dependent on the velocity of the lab in which the experimental apparatus was housed; we are thus presented with the idea that the measured speed of light shouldn't depend on the velocity of the measurement apparatus either. This is in contrast to the situation with wiggling ropes or pressure waves in air. 
A: After talking to several physics professors, reading various webpages, and thinking about Maxwell's equations, I think I have answers to my questions.
First of all, lots of people explained the answer to me using an LC (inductor-capacitor) circuit explanation (including a previous answer here), but I think it simply doesn't apply. LC circuits create oscillations that are often described with analogy to a pendulum, where the capacitor charge is analogous to the bob's position and the inductor magnetic field is analogous to the bob's momentum. This is a valid and useful analogy for an LC circuit. Here, the capacitor electric field is the restoring force and the inductor magnetic field is the inertia. However, importantly, these two fields are out of phase: the B-field is small when the E-field is big and vice versa. In contrast, the two fields are in phase for electromagnetic waves, showing that they are not LC circuits. Secondly, there is clear causation in an LC circuit where each field causes the other field, which is not so clear in electromagnetic waves.
Maxwell's equations in a vacuum are
$$\frac{\partial B}{\partial t} = -\nabla \times E$$
$$\frac{\partial E}{\partial t} = \frac{1}{\epsilon_0 \mu_0} \nabla \times B$$
In one extreme view, from Jefimenko, these equations do not express causation but are just a statement of fact. The laws of physics in our universe happen to state that E will be changing whenever there's a curl in B, or if you prefer, that there's a curl in B whenever E is changing. And vice versa with B changing and a curl in E. From this interpretation, it would not be legitimate to say that E is a restoring force for B or that B is a restoring force for E. Likewise, I don't believe that it would be legitimate to consider either of them as providing an inertial influence for the other in this view.
A different and more conventional view is that Maxwell's equations do show causation, with the curl of E causing B to change and the curl of B causing E to change. Here, each of them could be seen as providing the inertial influence for the other: e.g. when E is 0, there is still a curl in B, and this causes E to continue on past 0, and vice versa. Identifying the restoring force is a little less clear, but I think can be found by taking the curl the two Maxwell's equations shown and simplifying, which leads to
$$\frac{\partial^2 E}{\partial t^2} = \frac{1}{\mu_0 \epsilon_0}\nabla^2 E$$
$$\frac{\partial^2 B}{\partial t^2} = \frac{1}{\mu_0 \epsilon_0}\nabla^2 B$$
(see The Physics Hypertextbook). This shows that E "accelerates" due to its spatial curvature. Thus, I think it's at least somewhat reasonable to say that the spatial curvature (more correctly, the Laplacian) of E provides the restoring force for E, and the spatial curvature of B provides the restoring force for B.
Several people avoided answering my question by saying that light waves only make sense when using relativity. I disagree. It's certainly true that E-fields and B-fields are not distinct but are really two aspects of the same electromagnetic field (see Wikipedia Jefimenko's equations). However, this is nothing new, but is fully captured in Maxwell's equations; the coupling in them and the fact that they are always true, shows that E and B are inseparable from each other. Because it's in Maxwell's equations means that we don't need to consider further relativistic connections between the E and B fields to make sense of electromagnetic waves, despite the fact that these connections are interesting and useful in some situations.
Thus, my answer to my first question is Maybe. In Jefimenko's view of no causation, there is no restoring force or inertial influence for electromagnetic waves. These waves are fundamentally different. However, in the more conventional view of there is causation, then the spatial curvature of each field (the Laplacian) provides the restoring force for the same field, and the spatial gradient of each field (the curl) provides the inertial influence for the other field.
Regarding the speed of light, my understanding is that $\epsilon_0 \mu_0$ is just some given constant in our universe. There's no more making sense of why it has its value than there is of asking why the gravitational constant has its value. But, it does have a value and that sets the speed of light. It seems that it could be reasonable to consider this value as some sort of electromagnetic reactance of a relativitic aether that permeates our universe, or as simply a number that applies to the universe without an aether, which are fairly equivalent. By the way, I don't think it's legitimate to consider $\epsilon_0$ and $\mu_0$ separately because they only appear in Maxwell's equations in free space as a product. In other words, I think they are not independent parameters.
Thus, my answer to my second question is that the speed of light is set by the value of $\epsilon_0 \mu_0$, which simply happens to have some particular value in our universe. It's possible to imagine it being bigger or smaller, but that's not what it happens to be.
