Is a girl's reason for refusing a man's proposal—that he revealed the private matters of his ex-wife after their divorce—a convincing reason to reject him, even if this angers her mother?

Your marriage to this man is not religiously obligatory for you, nor is your obedience to your mother in accepting him. Ibn Taymiyyah said: "Parents are not to compel their child to marry someone they do not desire." Therefore, you will not be disobedient by refusing him, but strive to please your mother. If the man is religious and of good character, accept him as an act of filial piety towards your mother, for Allah may place blessing in this marriage because of this piety. The age difference does not prevent marriage. Inquire about him and seek guidance from Allah through Istikhara, and remember that his speaking ill of his previous wife is a religious error.