If they didn't "assert themselves," does that imply that they did exist? I think that this way of speaking is a little confusing, because we believe that current "laws" represent special cases of more general laws, rather than different laws entirely.

Massless particles inherently move at c. They can't be accelerated or decelerated because they have no inertia, although they do have momentum.

As far as we know, this was just as true right after the big bang. Particles, or field excitations or whatever, had no mass, and moved at c. They did have energy, and an energy density, and therefore were gravitationally attracted. This attraction would be described by quantum gravity, instead of General Relativity.

Once the particles acquired mass via the Higgs mechanism (probably at or about the same time that the modern-day forces became completely separated), the universe was still an ultra-hot quark plasma, so the newly massive particles still moved very rapidly. Just not at c.

Are you proposing that the laws change randomly or something?

If the laws of physics change with time, then what we thought were the laws aren't actually the laws, but rather the actual laws with parameterized time. It might make some experiments more difficult, but there is no philosophical conundrum. Actually, this idea is already implicit in lambda-CDM ("standard model" of cosmology), where there is a time-dependent "scale factor" in the Friedman equations.

Only matter and radiation must move at or below the speed of light. Relativity poses no limit on the relative velocities of objects, provided these velocities are acquired via the expansion of space. During the inflationary epoch after the big bang, space itself (probably) expanded at a rate faster than the speed of light. We think this process magnified small fluctuations, which nucleated the aggregation of matter into galaxies, that it separated different regions of the universe after they were in thermal equilibrium, and that it diluted away rare particles such as magnetic monopoles.

The whole idea of an "observable universe" is predicated on relative velocities greater than the speed of light. The event horizon at the edge of the universe exists because beyond that distance space (and the objects in it) are moving away from us at a speed greater than that of light, resulting in the causal separation of regions of the larger universe.

Actually, we know some rapid expansion must have occurred, because we can see objects which are apparently ~40 billion light-years away, even though the universe is only 13.8 billion years old.

The parent may have confused you with the choice of terminology. All particles, including the Higgs boson, are excitations of fields which permeate all spacetime. These fields have existed at least since the big bang. There is no need for any of them to have "come first."

It is true that today there are many sorts of massless particles, and it is very likely true that cooling after the big bang led to some rearrangements of the quantum fields (called the 'Higgs mechanism') which led to many particles acquiring mass or becoming more massive.

First, massless particles, like the photon or graviton, don't go past c. They go exactly c. Anything going faster would be a tachyon, which isn't like any of the massless particles we know. BTW, these BICEP2 results seem to confirm the existence of gravitational waves, and thus of gravitons. Otherwise, I would have said "supposed gravitons" or something like that.

There is also the concept of tachyonic fields, which are fields whose particles have imaginary mass. The Higgs before symmetry breaking occurred is an example, but despite their name, excitations of tachyonic fields propagate at or below the speed of light.

In short, you can't accelerate even massless particles past c. Nevertheless, objects can have faster-than-light relative velocities as long as these are acquired via the expansion of space.