Is it permissible to kill one's cousin (maternal aunt's son), or what is the legitimate religious punishment that can be carried out against him after he assisted the younger brother in stealing entrusted gold, taking into consideration the caution against legal accountability in a foreign country?

Willfully and wrongfully killing a soul is one of the gravest sins. The texts of the Qur'an and Sunnah contain severe warnings for those who commit it. Taking a life is not permissible except in one of three cases: the married adulterer, a life for a life, and one who abandons his religion and separates from the community. Even if a person deserves to be killed, the execution of the Hadd (prescribed punishment) is not the right of individuals but rather the right of the ruler or his deputy, such as a judge. Therefore, what you are contemplating is dangerous. Theft is a major sin that can only be proven by confession or just evidence, and it is the Islamic judiciary that determines its proof. You should report your cousin to the authorities, and if theft is proven against him, then those whose belongings were stolen have the right to reclaim them from him.