Thanks for your reply. As for studying under OPT, it is impossible? We can stay at home during OPT if job is not found, but we can't register for class?

What if I only take some courses as a non-matriculated student?

Thanks for your detailed explaination. I think I won't go to the school that doesn't sponsor F1. Currently, I am taking one course during F1. I am still doing thesis continuation, so my status is F1. They didn't ask me to show anything. I also asked for studying in OPT, they only need me to show I-94. Anyway, I will try to find another school.