We address the propagation of a single-photon pulse with two polarization components, i.e., a polarization qubit, in an inhomogeneously broadened "phaseonium" Λ-type three-level medium. We combine some of the nontrivial propagation effects characteristic for this kind of coherently prepared systems and the controlled reversible inhomogeneous broadening technique to propose several quantum information-processing applications, such as a protocol for polarization qubit filtering and sieving as well as a tunable polarization beam splitter. Moreover, we show that by imposing a spatial variation of the atomic coherence phase, an efficient quantum memory for the incident polarization qubit can be also implemented in Λ-type three-level systems.