Is a maiden's first marriage valid, and can she marry another without a divorce from her first husband, to whom she was secretly married?

If your cousin married you in the presence of two witnesses and a judge or a marriage officiant, then your marriage is valid, and divorce is obligatory before marrying another man. Marriage without a guardian is invalid according to the majority of scholars, but if a judge conducts it, it is not annulled, especially in countries where the Hanafi school of thought is prevalent. The marriage remains valid until the husband separates from his wife through divorce, khul‘ (divorce at the wife's instance), annulment, or death. It is sufficient for him to say to you, "You are divorced." If your 'iddah (waiting period—three menstrual cycles) ends, you become lawful for another man, with the necessity of officially registering the divorce.