In the isostructural oxides Ca3CoMO6 (M = Co, Rh, Ir), the CoMO6 chains made up of face-sharing CoO6 trigonal prisms and MO6 octahedra are separated by Ca atoms. We analyzed the magnetic and electronic properties of these oxides on the basis of density functional theory calculations including on-site repulsion and spin-orbit coupling, and examined the essential one-electron pictures hidden behind results of these calculations. Our analysis reveals an intimate interplay between Jahn-Teller instability, uniaxial magnetism, spin arrangement, metal-metal interaction, and spin-orbit coupling in governing the magnetic and electronic properties of these oxides. These oxides undergo a Jahn-Teller distortion, but their distortions are weak, so that their trigonal-prism Con+ (n =2, 3) ions still give rise to strong easy-axis anisotropy along the chain direction. As for the d-state split pattern of these ions, the electronic and magnetic properties of Ca3CoMO6 (M = Co, Rh, Ir) are consistent with d(0) < (d(2), d(-2)) < (d(1), d(-1)) but not with (d(2), d(-2)) < d(0) < (d(1), d(-1)). The trigonal-prism Co3+ ion in Ca3Co2O6 has the L = 2 configuration (d(0))(1)(d(2), d(-2))(3)(d(1), d(-1))(2) because of the metal-metal interaction between adjacent Co3+ ions in each Co2O6 chain, which is mediated by their z(2) orbitals, and the spin-orbit coupling of the trigonal-prism Co3+ ion. The spins in each Co2O6 chain of Ca3CoMO6 prefer the ferromagnetic arrangement for M = Co and Rh but the antiferromagnetic arrangement for M = Ir. The octahedral M4+ ion of Ca3CoMO6 has the (1a)(1)(1e)(4) configuration for M = Rh but the (1a)(2)(1e)(3) configuration for M = Ir, which arises from the difference in the spin-orbit coupling of the M4+ ions and the Co center dot center dot center dot M metal-metal interactions.