Is it permissible for a girl who is engaged by a valid Islamic contract to travel to stay with her fiancé and his mother in America until he finishes his studies, so as to help him avoid temptations, despite the absence of a stable place for them to live after marriage?

If the marriage contract has been concluded and the husband wishes the consummation (dukhulah) to take place abroad, there is no harm in that. In fact, it is commendable for the purpose of uniting the couple, maintaining chastity, and avoiding temptation, and this should be publicized. A wife traveling to her husband is better for both of them, and her travel must be with a mahram. However, if the intention is to travel to him without considering it a legitimate consummation, then this is frivolous and unacceptable to reason. If what is meant by engagement (khitbah) is merely a promise of marriage without a marriage contract, then travel is not permissible, and the rulings of marriage do not apply to it.