We present a study of the magnetic states and exchange coupling in transition-metal Mn, Fe, and Co atomic chains deposited on a self-corrugated Cu₃N-Cu(110) molecular network by means of first-principles calculations based on the density functional theory. The various adsorption sites on a bumping area of a self-corrugated Cu₃N layer are investigated where the atomic chains are formed at the initial stage of nanowire growth. We demonstrate, by calculating the ground-state magnetic configurations, that the exchange coupling, magnetic order, and anisotropies in atomic chains depend sensitively on their chemical composition and adsorption sites on the Cu₃N network. We find that the exchange interactions in atomic chains could lead to ferromagnetic or antiferromagnetic coupling of atomic spins depending on the position of the chain on the surface. The classical spin dynamics is investigated by means of the kinetic Monte Carlo method based on transition-state theory. Moreover we evaluate the Heisenberg-Dirac-Van Vleck quantum spin Hamiltonian for calculations of the magnetic susceptibility, in order to demonstrate the existence of quantum entanglement in the antiferromagnetic atomic chains at low temperatures.