Multiferroic biphase systems with robust ferromagnetic and ferroelectric response at room temperature would be ideally suitable for voltage-controlled nonvolatile memories. Understanding the role of strain and charges at interfaces is central for an accurate control of the ferroelectricity as well as of the ferromagnetism. In this paper, we probe the relationship between the strain and the ferromagnetic/ferroelectric properties in the layered CoFe2O4/BaTiO3 (CFO/BTO) model system. For this purpose, ultrathin epitaxial bilayers, ranging from highly strained to fully relaxed, were grown by molecular beam epitaxy on Nb:SrTiO3(001). The lattice characteristics, determined by X-ray diffraction, evidence a non-intuitive cross-correlation: the strain in the bottom BTO layer depends on the thickness of the top CFO layer and vice versa. Plastic deformation participates in the relaxation process through dislocations at both interfaces, revealed by electron microscopy. Importantly, the switching of the BTO ferroelectric polarization, probed by piezoresponse force microscopy, is found dependent on the CFO thickness: the larger is the latter, the easiest is the BTO switching. In the thinnest thickness regime, the tetragonality of BTO and CFO has a strong impact on the 3d electronic levels of the different cations, which were probed by X-ray linear dichroism. The quantitative determination of the nature and repartition of the magnetic ions in CFO, as well as of their magnetic moments, has been carried out by X-ray magnetic circular dichroism, with the support of multiplet calculations. While bulklike ferrimagnetism is found for 5-15 nm thick CFO layers with a magnetization resulting as expected from the Co2+ ions alone, important changes occur at the interface with BTO over a thickness of 2-3 nm because of the formation of Fe2+ and Co3+ ions. This oxidoreduction process at the interface has strong implications concerning the mechanisms of polarity compensation and coupling in multiferroic heterostructures.