Is a child raised by a woman from infancy until adulthood, but not breastfed by her, considered a محرم to her, with the rulings of a محرم applying to him, or is he a stranger to her, with the rulings of a foreign man applying to him, especially since her husband goes to work and leaves them alone in the house?

Sponsoring an orphan has a great reward, and there is indeed encouragement for it. If the aforementioned orphan is a non-mahram (definition) to the woman and there is no cause for mahramiyyah between them, and he has reached puberty, then she is obliged to wear full hijab (definition) in his presence. It is not permissible for her to seclude herself with him alone in the house in the absence of her husband and without the presence of a mahram, due to the prohibited seclusion involved.