Good question about the newborn babies dying, I've often wondered that myself. This has helped to understand it a little better. Although it's hard to get my mind around how one person's sin (Adam), can cause death to the rest of his posterity, even 'over those whose sinning was not like the transgression of Adam'.
"Therefore, just as sin came into the world through one man, and death through sin, and so death spread to all men because all sinned— for sin indeed was in the world before the law was given, but sin is not counted where there is no law. Yet death reigned from Adam to Moses, even over those whose sinning was not like the transgression of Adam, who was a type of the one who was to come." Romans 5:12-14