The effect of the background polarizability on the thermodynamic properties of a Coulomb crystal of ions is studied. The response of electrons is treated using the Thomas-Fermi (TF) and random phase approximations (RPA). For the case of ions fixed at their lattice sites, the energy of bcc and fcc crystals is calculated to first order in the screening parameter (kappa(TF)a)(2) (kappa(TF) is the TF wave number and a is the ion sphere radius). It is shown that in the RPA there exist domains of parameters (mass density rho and charge number Z) where energy of fcc crystal is lower than that of bcc. The effect of ion vibrations is studied using harmonic lattice approximation. It is shown that phonon modes are nearly identical in the RPA and in the TF approximation. The latter allows one to apply the Ewald technique to the construction of the dynamical matrix, which speeds up all calculations considerably. The main thermodynamic quantities of phonons are calculated as functions of the quantum parameter T(p)/T (where T(p) is the ion plasma temperature) and the screening parameter. The electron polarizability leads to a moderate increase of the phonon thermodynamic quantities as compared to the case of one-component plasma with rigid background (by approximately 30% at kappa(TF)a=0.8). Zero-point motion of ions modifies the aforementioned domains where fcc has lower energy than bcc for static ions. The effect is profound at small Z but leaves the domains unaltered at larger Z. The thermal vibrations of ions at T greater, similar T(p) eliminate completely the domains where fcc is thermodynamically preferable at T=0. The related model of Yukawa-Wigner solid is briefly studied. It is shown that neither bcc nor fcc crystal structures are stable in this model.