Is it permissible for the husband to live in his stepson’s apartment—the orphan’s apartment—which his father left for him, and sell his own apartment to expand his business, provided that he completely renovates the orphan’s apartment upon his marriage?

It is not permissible for you to reside in an orphan's apartment except by paying rent, even if the guardian gives permission. This is because the guardian's actions must be in the best interest of the orphan, and waiving the rent is not in their best interest.