Is it obligatory for the questioner and her mother to leave the house where the legitimate controls of mixing are not respected, and to move to another, safer house, given the presence of a non-mahram man who stays in the house and acts in a suspicious manner, the relatives' disregard for the controls of mixing, and the questioner's refusal to remove her niqab in front of her aunts' husbands?

Covering a woman's face from non-mahram men is obligatory according to the most preponderant scholarly opinions, and unveiling it before a non-mahram is a reprehensible act. Your cousin's husband should be advised gently and kindly regarding his reprehensible actions, such as entering upon women without permission and backbiting. If he does not repent, he should be shunned, provided that shunning him is likely to benefit him. If the house is constantly filled with reprehensible acts that cannot be prevented, and a safe alternative is available, it is obligatory to move to it. A woman living alone is permissible if there is no fear of temptation, but it is preferable for her to have someone else with her. Your cousin's daughter shouting at your mother is ill-mannered, and it is not permissible for her to shun you merely because you knocked on the door.