> [does gravitational wave detection imply the waves are particles]

No, not really.

However, if we deliberately quantize the gravitational waves, replacing the classical waves with large numbers of particles, the fact that the classical waves propagate at the speed of light means that the particles must be massless, like photons.

There are a couple of families of alternatives to General Relativity that have "massive gravity", which requires that gravitational waves (or the particles, if we quantize gravitational waves) have a small nonzero mass. This means light will propagate faster than gravitational waves, and with some care, this can solve a couple of the handful of problems about the very early hot dense universe, namely in how it became so thermalized across huge regions of space: any sort of pressure density released lots of photons which carried the local energy away faster than gravitation could "react" and contain the heat locally. Unfortunately for such theories, the mass of gravitational waves has to decay away well before the formation of the cosmic microwave background, since massive gravity would leave unmistakeable markings in it that we do not see.

Verifying that gravitational effects do not outrace optical effects effectively kills off all but the most complicated (early-decaying, thus dynamical) massive gravity theories, and those are rejected on parsimony grounds (the complexity is not needed to describe observations).

You're right that matter embedded in an inflating universe is not limited by the speed of light. As with galaxies now, the matter remains more-or-less at the same comoving coordinates throughout the history of the universe, even during inflation. We don't see shear effects in distant galaxies even given the redshift (if all Doppler/motion related) implies ultrarelativistic recession velocities. The galaxies far away look pretty normal, moving slowly within clusters, with the clusters separating. However, lots of new space is being created between each and every galaxy cluster, but the galaxies do not "feel" acceleration. They continue merrily along their almost exclusively timelike worldlines.

Inflation is essentially an extremization of this: lots of new space gets created in between the hot dense low-entropy matter of the pre-inflationary universe, and more new space gets created in less-dense/higher-entropy spots, which makes them even less dense / higher-entropy, which lets yet more space be created there, and so forth, for many "e-foldings". The pockets of gas do not "feel" like they are moving relative to one another, although if they had radars or the like, they could certianly see that during inflation they get much further away from each other (lots of radar signals will simply never return to the sender, or even be received at the target).

At some point in the inflationary epoch the production of new space "cools off" and inflation ends, after which the accelerating expansion is driven by the residual small positive cosmological constant. Structure formation then collapses the relatively dense relatively low entropy matter into the first stars and galaxies, while the space between these areas gets ever less dense and ever higher entropy. Some time after inflation we switch to the modernized version of the expanding Einstein-de Sitter universe (1932), which in the late 1980s/early 1990s was the starting point for the current standard cosmology (\Lambda-CDM). \Lambda-CDM does a much better job describing observations relating to the early universe (before the first galaxies) and recent-epoch acceleration than the Einstein-de Sitter model.

(Note, this is a broad stereotyping of cosmic inflation models; and for completeness, one can still get \Lambda-CDM from the hot dense phase (the "big bang") without inflation. Evidence from e.g. the Planck satellite has killed off some flavours of inflation and non-inflation proposals, and forthcoming evidence from Planck and other platforms will surely kill off many more.)