After reaching the age of 16, citizens have the right to change their last name on their own. Parental permission is not required.
According to paragraph 3 of Article 23 of the Law of Ukraine "On Mobilization Preparation and Mobilization", men who are dependent on three or more children under the age of 18 are entitled to a deferment. In fact, your husband currently supports two children. Pregnancy is not equated with the birth of a child, so your husband has legal restrictions on receiving a deferment and traveling abroad.
There are a number of legal grounds when one of the parents independently decides on the issue of the child's temporary departure abroad, including:
the place of residence of the child together with him/her has been determined by a court decision or confirmed by a conclusion of the guardianship and trusteeship authorities;
no measures of enforcement of the decision to establish visitation with the child and to eliminate obstacles to visitation with him/her are applied to him/her;
the period of stay abroad will not exceed one month;
the purpose of the trip must be treatment, education, the child's participation in children's competitions, festivals, scientific exhibitions, student olympiads and contests, the child's recovery and recreation abroad, including as part of an organized group of children, etc.;
informing the other parent, who does not evade and properly fulfills parental duties (if the place of residence is known), about the child's temporary departure outside Ukraine, indicating the purpose of the trip, place of stay and approximate time of return home;
existing arrears in alimony payments, the total amount of which exceeds the amount of the relevant payments for four months, confirmed by a certificate of arrears in alimony payments (issued by a state enforcement agency or a private enforcement agent).
According to Ukrainian legislation, a child under the age of 16 is allowed to travel abroad only when accompanied by an adult. Depending on the degree of kinship, a certain list of documents must be provided. Therefore, if a child is traveling with one of the parents or guardians, then in the conditions of martial law, which has been in effect since February 24, 2022, the notarized consent of the other parent is not required; it is enough to have the original birth certificate of the child with you.