Is it obligatory to inform the brother of what his wife stole from the elder brother's house, given his conviction that the plaintiff and her husband are not lying about her, along with the elder brother's pledge not to inform anyone of what his brother's wife did?

You are not permitted to reveal your brother's secret about his wife's theft, as that is considered a betrayal of trust, and revealing a secret is forbidden, especially if it causes harm. As for divorce, a promise of divorce is not a divorce. However, if he divorced her with an explicit and immediate pronouncement, then the divorce takes effect, and he has the right to take her back during her 'iddah, unless it was the third divorce.