What is the ruling of Islamic law regarding what the wife will take by force through American law, and is it considered permissible (halal) or forbidden (haram)? And what is the ruling on a religiously valid marriage (zawaj shar'i) after an American divorce?

If the husband utters the divorce or writes it with the intention of effecting it, the divorce occurs. Divorce issued merely by a ruling of secular courts is not effective. The divorced woman has specific rights according to Islamic law, and she is not entitled to more than what Islamic law has granted her, even if a court rules otherwise. As for the return of the wife after the third divorce, the statement is vague and requires consultation with trustworthy scholars to determine the validity of the remarriage.