Every human life requires a soul (of course), and although Satan can screw or rape as many women as he wants, God will only grant him the opportunity to procreate once using the soul God provides so that the Antichrist can be born. If there's miscarriage or the newborn dies at childbirth or later, Satan will once again be at risk of ending up in the Lake of Fire. However, God sees to it that that doesn't happen.
Connor's "cousins" are fully human, descended from real human parents from the beginning of the breeding program. Thereafter, the descendants are mostly a mix of boys and girls raised within the program, so each of them has a human soul. That continues for two millennia, ever since Satan read an early copy of Revelation (late first century). As you can imagine, he doesn't like the outcome, so he cooks up the wager with God, which requires that God provide a soul once Satan was ready to breed.
Satan kills one of the adults who was in the breeding program to get the guy's body and DNA, which Satan takes over (i.e. De Rosa's zombie body). All Satan has to do then is find some poor woman to serve as the AC's mother. De Rosa rapes her, mixing his DNA and hers. Her DNA is descended from someone who escaped the breeding program long ago, a lineage then lost to Satan. Her "wild" genes combined with De Rosa's genes from the breeding program is what gives Connor greater powers than those of his cousins and father. God provides Adam's soul at the moment of conception, and Connor is born.
De Rosa had no idea Campagna was a descendant of the breeding program. One sticky issue is why didn't Satan have Connor's genes looked at. Even though De Rosa knows this is the Antichrist, and safely assumes Connor is generally safe from an early death (the cousins all have powerful immune systems and are able to heal quickly from illness or injury), one would think he'd still be interesting in knowing how Connor's DNA differs from his own. Plot hole, for now.
In Catholicism, life begins at conception.
The fact that Satan chooses to rape Campagna, leading to the whole cascade of events in the trilogy, is pure coincidence, but I find those in interesting stories all the time. I think it's great when you can tie your characters to the heart of the plot. Connor's anger at the fact that Satan raped his mother leads to a cascade of events resulting in Satan's demise. After stabbing Satan with the dagger intended for Christ, Connor tells the dying Satan: I loved you, father, and I would torn the Earth in two for you, but you raped my mother for sport. ... Your evil acts created me, and in so doing, you sealed your fate.