Yes, your spouse and unmarried children under 21 years old may accompany you on H-4 visa and they may stay as long as you are on valid H-3 status.
No, your dependents may not work on H-4 visa. They have to apply for appropriate Work Visa.
Yes, your dependents may study on H-4 visa. They do not have to apply for separate F-1 student visa. They can study on the same dependent H-4 visa.
To apply for H-3 visa, your U.S. employer must submit Form I-129, Petition for Nonimmigrant Worker, with the regional USCIS center that has jurisdiction over the place where the training will be offered.
Note: The petition should not be filed more than six months prior to the commencement of the program.