Is it permissible to threaten to stir up trouble to prevent a sister from completing her studies far from home, especially when the parents approve of her travel?

It is not your right to threaten your family, nor does your nervousness or their distance from religion justify it. Your sister is permitted to study far from your region if the place is safe and she is accompanied by a mahram during travel; the presence of a mahram is not required during her stay. Ibn Baz said: "A woman's residence in a country without a mahram is harmless... as long as the work is among women and she is safeguarded from men." If there is a need to advise your family, let it be with gentleness and leniency. Allah says: "Invite to the way of your Lord with wisdom and good instruction," and the Prophet (peace be upon him) said: "Indeed, gentleness is not in anything except that it adorns it." You must focus on ridding yourself of nervousness by supplicating to Allah and adhering to Islamic manners when angry.