Yes, your dependents may study on J-1 visa. They don't have to apply for a separate student visa.
No, there are no travel restrictions on J-1 visa. You may plan with your employer and travel out of the U.S. any number of times while on valid J-1 visa.
The 'Two-Year Rule' is the common term used for a section of immigration law which requires that many exchange visitors return to their home country and be physically present there for at least two years after the conclusion of their exchange visit before they can be issued certain types of nonimmigrant visas, specifically H-1, L-1, K-1 and immigrant visas.