Is it considered a lie and haram for a wife to use a fake Facebook account and pretend to be a male "philanthropist" to communicate with a sheikh and with her husband, with the aim of convincing her husband to cooperate in resolving the issue of legitimate housing, and is this action correct?

If what you are doing falls under the category of using allusions (Tawriya) and indirect speech (Ma'areedh), and it harms no one, then it is neither forbidden nor a lie. The Sharia has permitted latitude in speech for the purpose of reconciliation, as stated in the Hadith: "The liar is not he who makes peace between people, saying good things and conveying good things." Lying has been permitted in three situations: war, reconciling between people, and a man's conversation with his wife and a woman's with her husband. Scholars have differed regarding the permissibility of outright lying in these cases; some have permitted it for the sake of benefit, while others have stipulated the use of allusions (Tawriya) and indirect speech (Ma'areedh) rather than outright lying.