You may start school with the F-1 pending since you legally entered in the US. If for some reason you are determined not to be eligible, you will be held to be illegally present back to the date your J-1 program ended (NOT when you applied, NOT the end of the thirty day travel period).
Yes, as you were told, you can go home and apply for the F-1 from there. While you are out of status the day the J-1 program ends, as long as you leave during the 30 day travel period (and don't work or otherwise violate the rules), it won't count against you going forward.