would the governor of hawaii born in japan to american parents be eligible to run for president?
What about an 80-year old man with only an elementary school education?
The United States Constitution requires that the President be a "natural born citizen". Whether natural born U.S. citizenship automatically implies native birth on U.S. territory, has never been determined definitively by a U.S. court. The first revision of the U.S. Naturalization Act however demonstrates the opinion of Congress in 1795 that citizens of the United States born abroad were not considered natural born.
So, it is not determined, we have the opinion of the 1795 congress, but that is not the same as in the Constitution.
One the second, yes.