A systematic electronic structure study of A₂FeReO₆ (A = Ba, Sr, and Ca) has been performed by employing the local-spin-density approximation (LSDA) and LSDA+U methods using the fully relativistic spin-polarized Dirac linear muffin-tin orbital band-structure method. We investigated the effects of the subtle interplay between spin-orbit coupling, electron correlations, and lattice distortion on the electronic structure of double perovskites. Ca₂FeReO₆ has a large distortion in the Fe-O-Re bond, and the electronic structure is mainly determined by electron correlations and lattice distortion. In the Ba-Sr-Ca row, the correlation effects at the Fe site are increased. The correlations at the Re site are small in the Ba- and Sr-based compounds but significant in Ca₂FeReO₆. Ca₂FeReO₆ behaves like an insulator only if considered with a relatively large value of Coulomb repulsion U_eff = 2.3 eV at the Re site in addition to U_eff = 3.1 eV at the Fe site. Ca₂FeReO₆ possesses a phase transition at 140 K where the metal-insulator transition (MIT) occurs between metallic high-temperature and insulating low-temperature phases. The spin and orbital magnetic moments are linear functions of temperature before and after the MIT but change abruptly at the point of the phase transition. From theoretically calculated magnetocrystalline anisotropy energy (MAE), we found that the easy axis of magnetization for the low-temperature phase is along the b direction, in agreement with experimental data. We found that the major contribution to the MAE is due to the orbital magnetic anisotropy at the Re site. X-ray-absorption spectra and x-ray magnetic circular dichroism at the Re, Fe, and Ba L_2,3 and Fe, Ca, and O K edges were investigated theoretically in the frame of the LSDA+U method. A qualitative explanation of the x-ray magnetic circular dichroism spectra shape is provided by an analysis of the corresponding selection rules, orbital character, and occupation numbers of individual orbitals. The calculated results are compared with available experimental data.