Can he file the I130 form for me and can I study while the petition is on process?

If you marry, he could file an I-130 but it would take awhile to be approved. So, you are right that you should maintain an underlying no immigrant status. Before you file an I-130, consider applying for an F-1 at a U.S. consulate as it is faster. A change of status to F-1 in the U.S. will take 8 months and will be denied if your J-1 expires in the interim. You also cannot attend classes until you have an F-1.

When you marry and your permanent resident husband will file I-130 petition for you, you will have to wait about 17 months for your priority date and up to a year longer to complete the process. During this time, you need some legal status to remain in the U.S. or have to return to your country and wait for an interview at the U.S. Embassy. If you change your status now to F-1, you will have the status necessary to remain in the U.S. while waiting for your green card. Remember, you cannot start classes until you get change of status approval, and you must attend school full-time and keep good grades - or your F-1 will will become invalid automatically. You also cannot work without permission of USCIS or change schools without observing the protocol with USCIS.