Is there a medium less dense than vacuum, in which light can travel faster than $c$? If not, can we make it?
The answer would seem to be "no", because you make a medium less dense by removing material from it. Once you get to a vacuum, you are only left with how good is the vacuum?
An experiment for an undergraduate optics lab would be to build a Michelson interferometer with a gas cell in one arm. As the gas is pumped out, the interference fringes shift. You can actually calculate the change in the effective speed of light at different air pressures, and project how it would change as the pressure declines further and further.
The limiting value is the speed of light in a vacuum.
It depends on the volume! You have to read something about the Casimir effect. Even in a complete vacuum, you always have virtual particles. Reality is quantum, and quantum vacuum is not empty! It cannot be!
So basically when the volume is bounded as in a capacitor, it seems there are fewer possible excitations inside (fewer kinds of virtual particles) than in a general big volume.
It looks like there's a lot of vacuums density possible, and some of them are emptier than others, even if all of them are empty!
The Casimir effect affects the force between capacitor plates and can be measured!
Is there a medium less dense than vacuum?
It depends from what you mean under vacuum. Let us take in account the local gravitational potential as a parameter of the vacuum. It is well known, that light follows the geodesic path of the space; near massive bodies the path of light is bent towards this body. But that is not the only phenomenon of the gravitational potential on light. Near massive bodies the light travels slower than farther away from those massive bodies.
If yes, light can travel faster than itself (as in vacuum) in that medium. If no, can we make it?
The speed of light is a local constant value and no light (or any matter) can travel in vacuum faster than c. What is said above about different c is valid only for a observer in a position with different gravitational potential. For example light travels - from our location in space - near black holes slower and in empty deep space faster c.
If you are talking about volume density, notice that "less density" does not imply a lower refractive index, although the correlation is often there. The Wikipedia page on the index gives some examples: https://en.wikipedia.org/wiki/Refractive_index#Density
I don't think it would make sense to assign any "density" to a vacuum, except for energy density, and I'm not aware of any absolute law relating any kind of density to the refractive index(except, of course, the optical density, of which it is a measure).
Now, it IS possible for the refractive index to be lower than one. The reason is that, when we talk about this index, we are talking about the change in phase speed, which can be greater than $c$ because it does not carry any information, and therefore there's no violation of relativity. Again, the Wikipedia page offers a good explanation and even some examples where the refractive index is less than one : https://en.wikipedia.org/wiki/Refractive_index#Refractive_index_below_unit
I've just read an answer to this inwhich the respondent objects that there is no more vacuumy a vaccum than total absence of matter, and that plots of the speed of light versus density extrapolated cross the zero density axis at the received value of $c$; but I presumed that the OP, in referring to beyond a vacuum meant beyond in the sense of virtual particles & vaccum energy & all that. If so ... it's conceivable that it would ... and yet that its doing so have no bearing on the possibility of any signal or substance travelling faster than this value. This kind of situation already exists in connection with radio waves & the ionosphere, in that the ionosphere technically has a refractive index to radio waves (in a certain frequency-band) less than unity. In such a medium, the speed of electromagnetic waves is greater than $c$ ... but - even theoretically - this superluminal speed of the underlying wave does not admit of any thing, in any sense in which "thing" can reasonably be said to have any meaning, being conveyed at superluminal speed. This is because any wave-packet composed of waves having any spread of wavelengths, howsoever slight that spread might be - even in the limit as the spread tends to zero, travels at the group velocity, which is always less than $c$. There is only meaning in the notion of conveying any substantial thing insofar as the wave varies - the notion of varying here comprising any stopping and-or starting of the signal. The wave speed, which has the superluminal size, only has this velocity insofar as it has infinite temporal extension both pastwards and futurewards, with which extension it is of a piece that there exists zero grounds for even so much as the notion of its speed being in any way immediately manifest - which is to say manifest actually as a speed . And yet it is manifest mediately, through the the bending of radio waves back earthwards at the boundary at which the ionosphere begins.
Basically, you find that whatever happens even in the realm of pure theory, the notion of any substantial or in any way discernible thing being conveyed at superluminal speed is always foiled - by reason of that very discernibility. And I am proposing it here that even if the answer to the OP's question could be reasonably maintained to be "yes", there would be a similar foiling of that notion of a piece with whatever reasoning it is by which it were so maintained.