2. Visa is different with your J1 status period. If your visa in passport still avoid to enter USA, you don't need new visa stamp. But if not, you need apply new visa stamp. How long is the new J1 visa stamp?
3. J2 can work, study.... after got the work permision. If you didn't, apply the work permision as soon as possible. In fact, J2 visa is much better then F1 if you could keep the J2 visa.