Yes, you may extend your stay on A-1 visa by a simple renewal of accreditation by the Department of State (DOS) pursuant to a request from a foreign government. Neither an application by the foreign national nor action by the USCIS is required.
No, you may not study or work on A-1 diplomatic visas. You must be coming to the U.S. to perform duties of an intrinsically governmental and non-commercial nature.
Yes, you may bring any member of your or your spouse's immediate family, or your close relatives to stay along with you.