In this paper we present an extensive study of the electronic, magnetic, and transport properties of finite and infinite periodic atomic chains composed of carbon atoms and 3d transition metal (TM) atoms using first-principles methods. Finite-size, linear molecules made of carbon atomic chains caped with TM atoms, i.e., TM- Cn -TM structures are stable and exhibit interesting magnetoresistive properties. The indirect exchange interaction of the two TM atoms through a spacer of n carbon atoms determines the type of the magnetic ground state of these structures. The n -dependent (n=1 to 7) variations of the ground state between ferromagnetic and antiferromagnetic spin configurations exhibit several distinct forms, including regular alternations for Ti, V, Mn, Cr, Fe, and Co, and irregular forms for Sc and Ni cases. We present a simple analytical model that can successfully simulate these variations, and the induced magnetic moments on the carbon atoms. Depending on the relative strengths of the carbon s, p and TM d orbital spin-dependent coupling and on the on-site energies of the TM atoms there induces long-range spin polarizations on the carbon atoms which mediate the exchange interaction. While periodically repeated TM- Cn atomic chains exhibit half-metallic properties with perfect spin polarization at the Fermi level, finite but asymmetric chains comprising single, double, and triple TM atoms display interesting spin-dependent features. These properties may be altered when these structures are coupled to electrodes. However, when connected to appropriate electrodes the TM- Cn -TM atomic chains act as molecular spin valves in their ferromagnetic states due to the large ratios of the conductance values for each spin type.