Yes, you may study on E-2 visa, however, you may not join a full length program like an F-1. You may take up a few credits at a university when they do not harm the primary interest of the visa.
Yes, you may bring your spouse and unmarried children under the age 21 years to stay along with you. They may stay in the U.S. as long as you maintain a valid E-2 status. You may also bring a domestic or personal servant on nonimmigrant status, provided you can show that:
Yes, your spouse may seek employment by applying for Employment Authorization using Form I-765, Application for Employment Authorization. However, your children are not allowed to work in the U.S.