Since Obama has been president for almost 2 full terms now, I'm going with the story that he is a natural born American born in Hawaii. There would be a tremendous constitutional crisis if somehow it could be proven at this time that he is not a natural born American, eligible to be president.
I have not read this whole thread, so someone may have pointed this out, but If Cruz who was born in Canada is constitutionally eligible to be elected president because his mother is an American citizen, then Obama is constitutionally eligible (and is) because his mother was an American citizen.