We determine the effects of film thickness, epitaxial strain and the nature of electrodes on ferroelectric phase transitions in ultrathin films of BaTiO₃ using a first-principles effective Hamiltonian in classical molecular dynamics simulations. We present results for polarization and dielectric properties as a function of temperature and epitaxial strain, leading to size-dependent temperature-strain phase diagram for the films sandwiched between `perfect' electrodes. In the presence of non-vanishing depolarization fields when non-ideal electrodes are used, we show that a stable stripe-domain phase is obtained at low temperatures. The electrostatic images in the presence of electrodes and their interaction with local dipoles in the film explain these observed phenomena.