No. At the time the Constitution was written, citizenship was conferred by the father. Obama would be a citizen the same as an anchor baby but not a "natural born citizen" as his father was Kenyan.
No, I don’t believe you’re correct. If Obama was born outside the U.S. and one of his parents was over 18 and a U.S. citizen he would be a natural born citizen. He could have been a dual citizen, but he still would be considered a natural born citizen.