J1 visa expiring and h1b applied.

Dear All,

I am currently holding a j1 visa for 12 month internship, that started on june 15, my visa expire may 30.
This means I can stay in US until june 30 for the 30 day grace period.

My company is applying for my H1b visa, and it was receive by USCIS on april 4, I do not have a case number yet.

My concern is that I believe I should not receive a an answer until october, so I am allowed to stay in the United states working for my company until the answer comes?

Do I need to fill out anything to protect me?

In addition to that, I received a letter from the DMV to update my status once my visa expires they need some sort of document to leave my drivers license valid.
Is there anything I can do about that?
Please advise.

Thank you so much