The work is devoted to the investigations of possible observational manifestations of protoobjects related to the "dark ages" epoch (10 < z < 1000), before formation of self-luminous galaxies and stars. These objects can distort the cosmic microwave background. Formation of these objects is described in the "pancake theory" and in the model of "hierarchic clustering." According to these theories we may consider these protoobjects as flat layers. We consider both Thomson (with Rayleigh phase matrix) and resonance (for complete frequency redistribution) scattering of cosmic microwave background radiation by a moving flat layer. The resulting anisotropy and polarization of cosmic microwave radiation are calculated for a wide range of layer optical thickness (from an optically thin layer to an optically thick one). Analytical solutions are also obtained for the case of an optically thin layer and are compared with the numerical ones.