Is it permissible to give zakat to a woman who swore not to receive help from her brother, after her brother stated otherwise, and is it permissible to give zakat to a person whose situation has not been verified based on the request of another person?

It is permissible to pay Zakat to a woman if it is highly probable that she is entitled to it, and her informing you of her poverty is sufficient. It is permissible to appoint a trustworthy person as an agent to pay Zakat. If a reliable person informs you that someone is entitled to Zakat, it is permissible to appoint him as an agent to deliver it to that person.