What is the ruling on a man's proposal of marriage to a woman, without having paid her dowry, then he dies? And does the woman inherit from him and observe the 'iddah (waiting period) for him?

If a valid Islamic marriage contract (nikah shar'i) has not been concluded between a man and a woman—meaning there was no offer from the guardian and acceptance from the husband, despite the fulfillment of conditions and absence of impediments—then the woman does not inherit from the man, nor is she required to observe an 'iddah (waiting period) or mourning, because she is not his wife but a stranger to him. If the family of the fiancée had received money from him, they must return it to his heirs.