The Program Designation State Department Branch in the Bureau of Education, Cultural Exchanges and Academic Programs (ECA) handles J1 visa extension, category changes, and program transfers. You will need to contact them directly if you have further questions about one of these issues. They can be reached at 202-401-9810.
Yes, you may study while on J-1 visa. However, unless you area J-1 student 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.