Physicochemical processes (H-atom sticking, diffusion, recombination, and the nuclear spin temperature of nascent H-2 molecules) important in the formation of molecular hydrogen have been experimentally investigated on amorphous solid water (ASW). A new type of experiment is performed to shed light on a longstanding dispute. The diffusion rate of H atom is directly measured at 8 K and is found to consist of a fast and a slow component due to the presence of at least two types of potential sites with the energy depths of similar to 20 and >50 meV, respectively. The fast diffusion at the shallow sites enables efficient H-2 formation on interstellar ice dust even at 8 K, while H atoms trapped in the deeper sites hardly migrate. The spin temperature of nascent H-2 formed by recombination on ASW has been obtained for the first time and is higher than approximately 200 K. After formation, H-2 molecules are trapped and their spin temperature decreases due to the conversion of spin states on ASW.