You can legally to stay after your F1 application is filed

before F1 is approved assuming you are not out-of-status at the time of filing 539. Your status during the period is F1 pending. I know that ex-B or F2 status holder are not allowed to attend school during F1-pending, ex-J2 H4, holder are allowed to attend school during F1-pending. I believe that you (H1 or ex-h1 holder) can attend school before F1 is approved, but not very sure. Ask school international office for that. However, you can stay during F1-pending.