No, they cannot work on the dependent A-3 status. They have to take appropriate work visa to work in the U.S.
Yes, they may study on the same dependent A-3 visa. They do not have to apply for a separate student visa.
Yes, you may leave and re-enter the U.S. on A-3 visa. You must have a passport valid for at least six months past the date of inspection and a valid visa each time you re-enter the U.S. Application may be made at any American Consulate. You should present a letter from your employer indicating that the employment will resume upon your return to the U.S. and a copy of the employer's I-94 or a letter indicating his/her continuing diplomatic service in the U.S.