Wrong! you cannot.

You cannot apply for the H1 just with no objection letter.

No objection letter came from Chinese government. If you get the letter from American government, you can start to apply for H1. But you must get the officer approval notice before you get your H1.

Extension J1 is pretty easy. You don't need to do anything. Only you need to do is tell your international office and get a new IAP66 form.