Is it forbidden to adopt and raise a child at home because he will eventually reach puberty while the wife is present, or is it permissible given that it is sponsoring an orphan, especially if that is the only solution to avoid divorce due to infertility?

Adoption is invalid in Islam, and a person attributing himself to other than his father is forbidden. You can raise one or more orphans without adoption, and if you want him to be a mahram (definition) to your wife, then her sister or mother should nurse him.