How can an orphan boy or girl be cared for within the home with one's own children, taking into account the legitimate Islamic guidelines when they reach maturity, especially if it is a girl who will become a non-mahram to the male children, and what is the virtue of sponsoring an orphan?

Islam encourages the sponsorship of orphans and promises abundant good for it. This is achieved by spending on the orphan, caring for them, and looking after their interests. It is not a condition for the orphan to live in the sponsor's home. It is possible to sponsor an orphaned girl without her being in the sponsor's home, and she can be breastfed to become a mahram.