No, to be eligible for J-1 student status, you must have funding, at least at first that comes substantially from any source other than personal or family funds, in the words of the regulation. Otherwise, to be eligible for J-1 status, you must participate in an exchange program formally established by a written agreement between governments, or between the University and a foreign government or institution. Thus if your funding comes from your own resources or from your family, and you will not be participating in an exchange program, you will be ineligible for J-1 status, and will have no choice but to come as an F-1.
Probably not. Some changes are prohibited by regulations, and others are possible only in very special circumstances. As an alternative to changing inside the U.S., it is sometimes possible to go home, obtain a new visa stamp in the other category, and reenter in the new status. Even if you go all the way home, though, the consulate may refuse to give you the category of visa that you want.