The CRBA would show is that he was eligible for a Congressional grant of citizenship which conferred the status “citizen”. He is a naturalized citizen.
Perhaps he believes that producing the CRBA with fanfare and saying “See? I told you I was a citizen” will end his problem. It might be an entertaining PR stunt but it does nothing to change his ineligibility for the presidency.
Yes. There is nothing that can change his ineligibility, period.
Yes. He could be setting this all up. Cruz is nothing if not theatrical. He is a citizen at birth by statute but not a NBC.