Fe/MgO-based magnetic tunnel junctions are among the most promising candidates for spintronic devices due to their high thermal stability and high tunneling magnetoresistance. Despite its apparent simplicity, the nature of the interactions between the Fe and MgO layers leads to complex finite-size effects and temperature-dependent magnetic properties which must be carefully controlled for practical applications. In this article, we investigate the electronic, structural, and magnetic properties of MgO/Fe/MgO sandwiches using first-principles calculations and atomistic spin modeling based on a fully parametrized spin Hamiltonian. We find a large contribution to the effective interfacial magnetic anisotropy from the two-ion exchange energy. Minimization of the total energy using atomistic simulations shows a surprising spin-spiral ground-state structure at the interface owing to frustrated ferromagnetic and antiferromagnetic interactions, leading to a reduced Curie temperature and strong layerwise temperature dependence of the magnetization. The different temperature dependences of the interface and bulklike layers results in an unexpected nonmonotonic temperature variation of the effective magnetic-anisotropy energy and temperature-induced spin-reorientation transition to an in-plane magnetization at low temperatures. Our results demonstrate the intrinsic physical complexity of the pure Fe/MgO interface and the role of elevated temperatures providing insight when interpreting experimental data of nanoscale magnetic tunnel junctions.