I haven't studied polarons in detail so can't comment on this, but they very probably relate to electron-phonon interactions. I have somewhat more to say about 2.:
Phonons are indeed bosons, and Goldstone bosons to boot, resulting from the spontaneous breaking of Galilean, translational, and rotational symmetries in the phase transition producing the solid. Three spontaneously broken symmetries $\implies$ three massless modes, or our one longitudinal and two transverse phonons. There is no direct correspondence between the broken symmetries and the phonon polarizations, however.
For many purposes the displacement field (of which phonons are the fundamental excitation) is well-approximated as a free scalar field, essentially when interactions between lattice atoms are harmonic, and there is no Higgs mechanism as such to speak of here.
But the analogy is not dead yet: anharmonicity in the interatomic potential gives rise to cubic and quartic phonon interactions which are known to be relevant to some phonon-phonon scattering processes. Whether or not this produces an interaction analogous to the Higgs mechanism's quartic coupling clearly depends on the sign and form of these quartic and higher-order anharmonic phonon-phonon interactions, which I understand to be a primarily experimental problem, and will depend very strongly on the material.
You would also need a quadratic phonon interaction to reproduce the characteristic Mexican hat potential, but this can't be done with anharmonicity (quadratic corrections to a harmonic interatomic potential are obviously not anharmonicities) as this will simply change the phonon dispersion. There may be some mechanism that produces quadratic phonon-phonon interactions, but I'm not aware of any. At any rate, this is not essential to produce an interaction with nontrivial minima, which is what is really wanted, so this is a bit of a technicality.
Finally, Phonons of various polarizations are often described as having an effective mass around certain critical points in the Brillouin zone, and can gain it around $\Gamma$ due to interactions with other excitations, but neither of these directly relate to your question about analogies with the Higgs mechanism, which I would understand as phonons gaining mass through a symmetry breaking self-interaction.