Up to now, I haven't been able to find an answer for a big question:
can a foreign-born become US president?
In the US Constitution, it's written that no foreign-born is "eligible" to the presidential office.
The actual restriction is that the President and Vice President be
natural born citizens, which means that they are a citizen by the nature of the birth. This would include persons born in the United States or its territories, or persons born outside the country to two US citizen parents, or one parent who has lived in the United States for a sufficient period after a certain age (I think it is something like 4 years after the age of 14).
There have been proposals that the Constitution be amended to permit naturalized citizens to become president, but have gone very far, perhaps because their is a sense that it would be for the benefit of particular individuals such as Schwartzenegger at present, or Henry Kissinger in the past. One website promoting such an amendment also mentions the beneficial effect it would have for Jennifer Granholm, the Canadian-born governor of Michigan, in order to make the issue seem more bipartisan.
For an interpreted version of the Constitutioon you might want to check the following.
Constitution of the United States 
