What is the ruling on raising an orphaned male child in one's home when the family is unable to conceive, and is the wife obligated to cover her hair from him upon his reaching puberty?

If sponsorship means taking care of the orphan and managing his affairs without attributing him to one's lineage, then this is a good deed, and the sponsor of the orphan will be with the Prophet, peace and blessings be upon him, in Paradise. As for the ruling on a wife uncovering her hair in front of an orphan child after he reaches puberty, he is a stranger to her, and she must observe full hijab (definition) from him and not be alone with him. The same applies to all women in the family. This awkwardness can be overcome by giving the orphan child valid suckling before he reaches two years of age, from one of the wife's sisters, or her mother, or another wife of the husband, or his brother's wife, so that he becomes a mahram (definition) to her.