LiₓCoO₂ is a prototypical layered cathode material for Li-ion batteries, yet its accurate description from first principles remains challenging because of self-interaction errors, weak interlayer interactions, and a complex magnetic energy landscape. Here, we present a systematic investigation of the structural, electronic, magnetic, and electrochemical properties of LiₓCoO₂ (x=0,1) using density-functional theory augmented with self-consistent Hubbard corrections and long-range van der Waals interactions, together with a systematic exploration of possible magnetic states. The on-site interactions on Co-3d and O-2p states, as well as inter-site Co-O interactions, are determined from first principles using linear-response theory in the framework of density-functional perturbation theory, with Löwdin-orthogonalized atomic orbitals employed as Hubbard projectors. For LiCoO₂, the inclusion of Hubbard corrections provides an accurate description of the structural properties, while the electronic structure is very sensitive to the choice of Hubbard projectors. In particular, frontier Wannier-function projectors substantially improve the description of the occupied electronic states compared with localized atomic orbitals. For CoO₂, we demonstrate that a systematic exploration of the magnetic energy landscape is essential to identify the lowest-energy low-spin ground state. However, the resulting Hubbard-corrected electronic structure is insulating, consistent with the prediction of the HSE06 hybrid functional, but in contrast to the experimentally observed metallic behavior. Structural relaxation further drives the system toward a different metallic solution with an electronic configuration inconsistent with low-spin Co⁴⁺ character. Despite these limitations, the calculated intercalation voltages agree well with experiment, with deviations as small as 2%.