The structural, electronic, and phonon properties of the BaTiO3(0 0 1) surface and the Fe/BaTiO3(0 0 1) interface have been studied within the density functional theory. Attention is paid to the lattice instabilities (soft phonon modes) that induce ferroelectric distortions in the surface and the interface. A phonon-induced monoclinic (Cm) thin-film counterpart of the low-temperature rhombohedral (R3m) ferroelectric bulk BaTiO3 phase is found. The changes in crystal structure, electronic density of states, atomic charges, and magnetic moments associated with the ferroelectric distortions are discussed comparing the results of the standard GGA and the hybrid DFT calculations. The magnetoelectric coupling at the Fe/BaTiO3(0 0 1) interface is investigated by the analysis of changes in magnetic moments on Fe and Ti atoms induced by the atomic displacements perpendicular and parallel to the surface.