We report on the electronic structure and magnetic properties of hybrid heterostructures of the ferromagnetic double perovskite Sr₂CrReO₆ (SCRO) and ferroelectric BaTiO₃ (BTO) calculated in the GGA approach using the fully relativistic spin-polarized Dirac LMTO method. The electronic band structure and optical properties are studied in the BTO and SCRO oxides as well as in SCRO/BTO heterostructures with different supercells: SCRO/BTO monolayered (1 x 1 x 1), (2 x 2 x 1), and (1 x 1 x 2) heterostructures. We investigated theoretically the magnetization, spin and orbital magnetic moments as well as magnetocrystalline anisotropy energy (MAE) in the SCRO on the BTO substrate as a function of the temperature assuming that the finite temperature can be mimicked by the experimental lattice constants corresponding to this temperature. We found that the spin and orbital magnetic moments are monotonic, almost linear functions of temperature but change abruptly at the points of the BTO structural phase transitions. The magnetization M also possesses "jumps" at the structural phase transitions in agreement with the experiment. However, the theoretically predicted jumps in the total magnetic moment are much smaller than the experimentally measured ones. We found that the Cr/Re disorder increases the jump in the magnetic moment by an order of magnitude compared to the calculated data without disorder. The microscopic origin of such huge effect is discussed. The theoretically calculated temperature dependence of the total magnetic moment with 25% of the Cr/Re disorder is in excellent agreement with the critical amplitude, obtained from the fitting of the experimentally measured magnetization M(T). From the theoretically calculated MAE we found that the easy axis of magnetization for the tetragonal and orthorhombic phases is along the 〈001〉 direction in agreement with the experiment. However, it changes to the 〈11[1]〉 direction in the rhombohedral phase. The intersite Cr/Re disorder reduces the MAE for the tetragonal and orthorhombic phases and increases it for the rhombohedral phase. We found that the major contribution to the MAE is due to the orbital magnetic anisotropy at the Re site. We also found that the MAE in the monolayered SCRO/BTO heterostructure is significantly increased in comparison with the MAE in SCRO on BTO substrate. The element-specific x-ray absorption spectra as well as the x-raymagnetic circular dichroism at the Ti, Ba, and Re L₂,3 edges in SCRO/BTO heterostructures are investigated theoretically from first principles and compared with available experimental spectra.