Is the speed of light in a vacuum constant? Why/why not? If not, how can one vary it?
Yes. The speed of light in vacuum is one of Nature's constants, and is a cornerstone of modern physics (see above). This is the basic assumption of Einstein's theory of relativity. This theory is both mathematically consistent, and had been tested innumerous times, and is therefore accepted as correct.
Light, though, can be slowed down in transparent media such as air, water and glass. The ratio by which it is slowed is called the refractive index of the medium and is greater than one. This was discovered by Jean Foucault in 1850.
Interestingly, when dealing with finite (as opposed to infinite) light beams, it was shown that even in vacuum, single photons travel slower than the speed of light, due to an internal change in the beam's structure. http://arxiv.org/abs/1411.3987. Thus, in fact, the speed of light should be considered as an absolute upper limit.