What is the ruling on a sister remaining in America alone to study without a mahram, and is the brother sinful if he leaves her and returns to his country, given the father's knowledge and approval of her staying?

A woman's residency does not require the presence of a mahram. However, residing in those countries is fraught with dangers and risks, especially for a young woman. You should not leave your sister alone there, and you must advise her to return with you or to wait until she finishes her legitimate studies. If you leave her, there is no sin upon you if you advised her and admonished her, and she did not respond to you; for every soul is responsible for its own actions.