Neutron stars are sources of strongly polarized emission in X-rays or soft gamma-rays due to the presence of strong magnetic fields. Radiation transport of soft X-rays in neutron star surface layers is critical to the determination of the emergent anisotropy of light intensity and polarization signatures. Additionally high-energy photons propagating in neutron star magnetospheres can be attenuated by QED processes like photon splitting and magnetic pair creation. In this thesis, I explore the scattering transport in the classical magnetic Thomson domain using Monte Carlo technique. Representative results for emergent polarization signals from surface layers are presented for both localized and extended surface regions with magnetic field strengths that are of broad applicability to different neutron star classes. These results provide an important background for observations acquired by polarimetry missions like IXPE. I also explore polarization-dependent opacities for the two QED processes in static dipolar or twisted magnetospheres of highly magnetized neutron stars like magnetars, calculating attenuation lengths and determining escape energies, which are the maximum photon energies for transparency out to infinity. These opacity calculations put constraints on the possible emission locales and the strengths of the magnetospheric twists, and apply not only to magnetar flares but also to their quiescent hard X-ray tail emission. An exploration of photon splitting attenuation in the context of a resonant inverse Compton scattering model for the hard X-ray tails derives distinctive phase-resolved spectroscopic and polarimetric signatures, of significant interest for future MeV-band missions such as AMEGO and e-ASTROGAM.