Concerning Maria's firstborn, I believe that he was actually Irregular, and that the Crim line was born from Adam's unnamed father (assuming he was born by normal means, which isn't stated so that's up for debate.) I think that that's why Irregular appears in MotC's womb once Eve is no longer in the doll, and why Hansel explains that "someone else" is in there when Irregular attempts to grab the doll. Maria is the only entity that can birth Irregular <--My personal belief.
Concerning the "Soul of Adam"(SoA) references throughout the years, I think that it's more figurative than literal. Also, the theory that the SoA are his reincarnations seems to imply that he only reincarnates in those who share his bloodline, which hasn't been (proven) to be the case with H&G, the paragons of reincarnation. I'm not saying that he can't be reincarnating as such, just that I find it unlikely.
Instead, I think that a SoA is someone who parallels the role that Adam played in the world during his lifetime. The main person that comes to mind in this assumption is Kyle; he parallels Adam in that his actions resulted in an insanity in a woman that loved him, which in turn resulted in a great tragedy. Adam/Eve resulted in the Original Sin, Kyle/Riliane resulted in Genocide. Another possible example is Gallerian/Nemesis resulting in World Destruction, though this is a different kind of love (father/daughter as opposed to romantic love) and I don't think that Gallerian was ever actually referred to as the SoA.) The parallel is still there, however, and Gear, the current SoA, does seem to have some sort of connection to Gallerian.