What is the Church’s official teaching about Salvation for unbaptized souls? For example, what about people born into the Buddhist or Hindu or Islamic religions? Or what about a child lost to a miscarriage?
For the Church’s teaching on the salvation of non-Christians, see paragraphs 839-848 of the Catechism of the Catholic Church, especially this section:
"This affirmation [no salvation outside the Church] is not aimed at those who, through no fault of their own, do not know Christ and his Church:
"Those who, through no fault of their own, do not know the Gospel of Christ or his Church, but who nevertheless seek God with a sincere heart, and, moved by grace, try in their actions to do his will as they know it through the dictates of their conscience – those too may achieve eternal salvation.
“Although in ways known to himself God can lead those who, through no fault of their own, are ignorant of the Gospel, to that faith without which it is impossible to please him, the Church still has the obligation and also the sacred right to evangelize all men” (CCC 847-848).
For the Church’s teaching on the fate of unbaptized babies, please see paragraph 1261 of the CCC:
“As regards children who have died without Baptism, the Church can only entrust them to the mercy of God, as she does in her funeral rites for them. Indeed, the great mercy of God who desires that all men should be saved, and Jesus’ tenderness toward children which caused him to say: ‘Let the children come to me, do not hinder them,’ allow us to hope that there is a way of salvation for children who have died without Baptism. All the more urgent is the Church’s call not to prevent little children coming to Christ through the gift of holy Baptism.”
For an overview of the historical development of the Church’s teaching on the doctrine extra ecclesiam nulla salus (Latin, “Outside the Church, no salvation”), I recommend the article “Can Outsiders Be Insiders” by Fr. Peter Stravinskas: