  We provide a comprehensive solution to the lattice dynamics problem in the
two dimensional Holstein model at finite electron density and finite
temperature. We work in the physically relevant adiabatic regime and vary the
electron-phonon interaction from the weak coupling perturbative window to the
strong coupling polaronic regime. We explore three typical electron densities,
dilute - where spatial correlations between polarons is weak, intermediate -
where correlations are significant, and half-filling - where there is long
range checkerboard order at low temperature. We use two methods both of which
exploit the "slowness" of the phonons to handle the problem. These are (i)~a
standard random phase approximation (RPA), adapted to capture small quantum
fluctuations on Monte Carlo generated classical thermal backgrounds, and (ii)~a
Langevin dynamics scheme, with a simplified ``thermal noise'', that can address
large amplitude dynamical fluctuations. The Langevin scheme, as we argue in the
paper, is the superior method in the strong coupling part of the phase diagram,
where lattice distortions are large. It reveals a non trivial multi-peak
momentum resolved spectrum with a high energy part, on the scale of the bare
phonon frequency $\Omega$, and a low energy peak at $\omega \ll \Omega$. Below
the polaronic threshold, the high energy dispersion changes only modestly with
temperature $T$, while the broadening, arising from mode coupling, increases
linearly with $T$ at low temperature. The low energy peak shows up at strong
coupling and finite temperature and arises from the slow tunneling of polarons.
The tunneling events become spatially correlated as electron density increases
towards half-filling, and the weight becomes strongly momentum and temperature
dependent. We suggest the analytic basis of these results.
