Is divorcing a wife for the the third time considered valid if two previous verbal divorces were documented in court? And is marrying an American woman for the sake of residency and benefit forbidden?

Residing in non-Muslim countries entails risks, and scholars have been stringent regarding it, setting conditions for its permissibility. It is not permissible to issue a false certificate of being unmarried, as this is a false statement and false testimony, which is forbidden and a grave major sin. Jurists have differed on whether a divorce occurs when a man falsely claims he has divorced his wife, and the preponderant opinion is that it is not considered a divorce. It is permissible to marry a righteous Muslim woman of any nationality, and it is permissible to marry a chaste woman from the People of the Book, though it is not recommended. The interest mentioned in the question (marriage for the sake of residency) involves religious prohibitions.