Is a wife riding in a car with her husband's brother (who is the age of her father) or with a taxi driver, to transport her and the children to some place, considered an unlawful seclusion (khalwa)?

The husband's brother is a stranger (non-mahram), and it is not permissible to be in seclusion with him. Some scholars have gone as far as to consider a woman riding alone with a stranger in a car as impermissible seclusion (khalwa). Therefore, it is not permissible for your wife to ride with your brother in a car alone, or with the presence of indiscriminating children. However, if she rides with him in the presence of discriminating children, there is no objection, provided that temptation is safeguarded against. Similarly, regarding her riding in a taxi with a non-mahram driver, it is not permissible for her to be alone with him. But if she is accompanied by other women or discriminating children, it is permissible, provided that temptation is safeguarded against.