Diva, your question is bit tricky. I asked with different people and I got different answers. Even immigration lawyers are also not clear about this. As I am also F1 Student, I have gathered few information about your question. As per rule, F1 status should be invalid for the period granted under the TPS. If you wish to travel home and come back, you have to apply for the I-131 form, while I-20 will not work. On the other hand, you can maintain your F1 status if you wish to, which wont hurt you and help you in the future for visa extensions. But you can also take a break in your studies and reinstate to the same F1 category before TPS expiration period by filing I-539 form. Since H1B petition has been complicated and in lottery system, rather you can be sponsored through your company for green card with the use of EAD(http://www.uscis.gov/green-card/green-card-through-job/green-card-through-job-offer). Try to find out the good company and work hard for them so they can sponsor you in the future. America is not like before and it seems to be quiet difficult to gain Permanent residency through our study unlike other countries (Australia, New zealand, Canada). Decision is on your hard. Best of luck...