We solve exactly the one-dimensional diffusion-limited single-species coagulation process (A+A -> A) with back reactions (A -> A+A) and/or a steady input of particles (B -> A). The exact solution yields not only the steady-state concentration of particles, but also the exact time-dependent concentration as well as the time-dependent probability distribution for the distance between neighboring particles, i.e., the interparticle distribution function (IPDF). The concentration for this diffusion-limited reaction process does not obey the classical "mean-field" rate equation. Rather, the kinetics is described by a finite set of ordinary differential equations only in particular cases, with no such description holding in general. The reaction kinetics is linked to the spatial distribution of particles as reflected in the IPDFs. The spatial distribution of particles is totally random, i.e., the maximum entropy distribution, only in the steady state of the strictly reversible process A + A -> A, a true equilibrium state with detailed balance. Away from this equilibrium state the particles display a static or dynamic self-organization imposed by the nonequilibrium reactions. The strictly reversible process also exhibits a sharp transition in its relaxation dynamics when switching between equilibria of different values of the system parameters. When the system parameters are suddenly changed so that the new equilibrium concentration is greater than exactly twice the old equilibrium concentration, the exponential relaxation time depends on the initial concentration.