We report measurement of room-temperature compressibility, thermal expansion, thermoelectric power a(T) at various pressures P < 20 kbar, basal-plane resistivity rab (T), magnetic susceptibility and thermal conductivity k(T) taken on single-crystal or cold-pressed Na0.57CoO2. An enhancement of a large thermopower with a change of slope occurs on heating near 100 K, but this enhancement is progressively suppressed by pressure. The c-axis thermal expansion is large in the interval 150 K <T < 250 K where the c-axis resistivity exhibits a smooth transition from a metallic to a non-metallic temperature dependence; but the basal-plane thermal expansion remains negligible for all temperatures T < 300 K. On the other hand, the basal-plane room-temperature compressibility is large in the interval 0 < P < 22 kbar, becoming negligible in the range 22 < P < 45 kbar, whereas the c-axis room-temperature compressibility is anomalously large in the pressure range 22 < P < 35 kbar. The basal plane resistivity is prop. to T^(3/2) below 175 K where there is 3D metallic conduction; it rises less rapidly with temperature where the metallic conduction is confined to 2D. The phonon contribution to the thermal conductivity of a cold-pressed ceramic sample is not suppressed, as previously reported. These findings are rationalized with the aid of the virial theorem, recognition of a pinning of the nominal Co(IV)/Co(III) redox couple at the top of the O2-:2p6 bands, and a schematic location of the a1T and eT antibonding bands of this couple with respect to the Fermi energy.