First-principles calculations in the framework of the generalized gradient approximation together with U on-site Coulomb corrections in the GGA + U approach to density functional theory (DFT) are performed to investigate the structural stability, magnetic order, and magnetocrystalline anisotropy of one-dimensional (1D) cobalt-oxide chains on Rh(553) step-surfaces. We found that the chains' magnetic and structural stability strongly depends on the oxygen concentration, η. It is determined that there exist competing direct ferromagnetic and indirect antiferromagnetic exchange interactions in the dopedoxygen 1D linear chains, and in general, the oxygen doping stabilizes the antiferromagnetism. For pure Co linear chains and low oxygen concentrations, in which η ≤ 0.1 monolayers (ML), the ferromagnetic solution is the ground-state magnetic configuration. For η > 0.2 ML, the antiferromagnetic arrangement stabilizes through superexchange interactions. The strong influence regarding the oxygen-doping on the Co linear chains' structural properties is evidenced when a small dimerization between the Co atoms at low O concentrations emerges. In contrast, dimerization in the Co chains is suppressed when the system is "oxygen-free" or η > 0.2 ML. The increase of oxygen concentration strengthens the pd hybridization between Co d-states and O p-states, leading to an electronic redistribution of the majority and minority bands of the Co d-states. Such a redistribution yields to the formation of more localized bands. A significant reduction of the local magnetic moment in the Co atoms is followed. The robustness of the DFT + U results is also discussed to some extent. Throughout a perturbative analysis, we also investigate the oxygen dependence on the magnetocrystalline anisotropy energy (MAE) for the Co-oxide chains, which ranges from 0.4 to 1.2 meV. Interestingly, magnetization directions canted to the wires' direction or perpendicular to the Rh terrace are determined. Their origins are discussed in terms of the local contributions to the MAE.