I am replying at the level of a highschool student.
I will use your last comment:
The fact that c is the speed of SOMETHING is derived from working through the basic units of the equation e=mc^2, until you are left with c being distance over time, once all the other units are cancelled out.
This is a complicated way of looking at a dated equation, which is no longer used in particle physics since it is not a fundamental property but result of the mathematics. Fundamental are the four vectors in the pseudoeuclidean space of special relativity, space and time and energy and momentum. The inertial form can be derived from these simple assumptions
That is speed, but does it HAVE to be the speed of light? That it is the same value as the speed of light in a vacuum could be just an artifact.
The speed calculated from E=mc^2 is for an effective inertial mass of a massive macroscopic body. It is useful to show how a massive body behaves under large accelerations, but irrelevant for basic physics. The speed of light in vacuum is an axiom of special relativity, and special relativity has been validated innumerable times in nuclear and particle physics.
Does the Higgs field have a 'speed', or is it absolutely stationary?
Fields are mathematical constructs that fill up all space, all the elementary particles have a field over all space. A photon field, a neutrino field etc are hypothesized mathematically over all space. It is the creation and annihilation operators acting on these fields that have measurable meaning. Higgs bosons appear with the creation operators on the Higgs field. Fields are stationary by construction, like extra coordinate systems.
If there were no Higgs boson, EVERYTHING would travel at the speed of c.
If there were no Higgs boson a new theory would be needed. The fact that the Higgs boson has been found at LHC is a validation of the present standard model of particle physics.
At the universe at present the Higgs field gives mass only to the elementary particles of the table. The atoms and molecules we observe would still have an invariant mass due to the strong force in the nucleus and the rules of special relativity. Whether bound states would form is a moot question.
The only laboratory we have is the cosmological history of the universe, where due to the recovery of the symmetry, which is broken at the low energies we live in, the elementary particles have zero mass up to ~10^-12 seconds after the big bang. The particles bouncing around there still have invariant masses when in pairs and systems.
To answer your question, no the Higgs field cannot be manipulated in our present universe, and it certainly has nothing to do with the speed of light. In this sense the symbol "m" is miss used in E=mc^2, it is a variable, it is just a mathematical result of the special relativity equations, which do assign an invariant mass to each four vector and thus to each particle.