Is Prince Hashim of Jordan eligible to run for President of the United States?

Queen Noor (Hashim's mother) renounced her American citizenship when she married King Hussein of Jordan, so the straight answer to the question is no, Hashim is not eligible to run for President of the United States.

What world do we live in where an American woman can not travel and have her baby in another country because that baby won't be "american" yet people from other countries can come here and have a baby and that baby is American?

A book the Founding Fathers read at the Continental Congress was "The Laws of Nations," by Emmerich de Vattel. In it, we find, "Natural-born citizens are those born on the soil, to parents who are citizens". Why "parents" and not "parent"? Because if a child had a foreign parent that child would be eligible for dual citizenship--which the founders didn't want. They were so against dual citizenship that they banned it outright. (It was not made legal in the US until 1967.) They expressly said that the reason they placed the "natural-born citizen" provision in the Constitution was to prevent dual citizens from being President. Ted Cruz was born with dual citizenship to three different nations: Canada, the US and Cuba. Ted likes to emphasize the fact that he has US citizenship; but that's not what disqualifies him. It's his Canadian and Cuban citizenship which disqualify him. Dual citizens cannot be President. This is not controversial. This was the position in the US for the first 240 of its existence--which is why you can search the "birthplaces" of all previous presidents and not find "Canada" as a birthplace. Or "China" Or Nicaragua.