Is what the brothers are doing—letting their sister travel alone a distance of 1000 km and stay for three days without a mahram, despite the questioner's objection—considered dayyoothah (cuckoldry/lack of protective jealousy), and would the questioner be like them?

The majority of scholars hold that a woman's travel without a mahram (definition) is not permissible except in cases of necessity, and her travel for training courses is not considered a necessity. A woman's work in a mixed-gender environment is a corruption. The sister who violates Islamic law should be advised gently and wisely, and she should be informed of the rulings of Islamic law regarding watching TV series, musical instruments, and birthday celebrations. If she insists on the violation, it is permissible to shun her as a deterrent, while considering the overall benefit. It is incumbent upon brothers to enjoin their sister to do good and forbid her from evil; their negligence in this is a sin. However, one is not considered a dayyuth (pimp/cuckold) unless he condones his family members committing immorality.