I am not sure, but i think that it's possible.
NO. I-20 is the document issued by the school to certify that you have enrolled in certain course at certain period of time. Usually it expires at the end of the course and if you need to continue your study, you have to enroll for new course.
I-20 is not a visa, but you need to submit one to USCIS if you are applying for student visa.