Answer #1 No.
The Constitution contains the term "natural born" for a US president. This term was used to prevent anyone with the possibility of having a foreign allegiance from becoming commander-in-chief of US forces, for obvious reasons (super fifth column, anyone?) A person with dual citizenship owes allegiance to both the US and the foreign government. They are required to obey the laws of both countries, which of course for a sitting US President would be disastrous to the US nation.
The term "natural born" is used to mean a singular allegiance to one and only one nation. Dual citizenship contradicts this.
(Note that it is unique that the US president is also the commander-in-chief, which entrusts the office of president with great power, so a concern of allegiance was on the minds of the framers of the Constitution. Senators and Representatives can be naturalized; the President cannot, because of the Commander-In-Chief power.)
All NATURALIZED citizens have to take an oath of ALLEGIANCE, to confirm they only have allegiance to the US and no other country. But this is not enough to fully repudiate possible allegiances to other countries.
So "allegiance" is the key here -- at the time of the writing of the US Constitution, "natural born" was a term carefully chosen to define that allegiance - it means having a singular allegiance that derives from one's having been born of the soil of the nation - in this case, the US nation. This results in a kind of "super-allegiance".
That is what the framers meant, and Constitutional lawyers have always recognized this to be the meaning.
So according to the Constitution, which explicitly and unambiguously uses with the term "natural born" when referring to qualifications for someone to be the US president, no, a US president cannot have dual citizenship. They must have that singular allegiance that can only be derived from having been born of US soil - "born here".
Answer #2 Yes. But they shouldn't. The Constitution does not say anything to that effect, but I agree that a US president shouldn't, because he would owe allegiance to two countries. Would it not be a conflict of interests? ____________ Answer #3 Yes. And why not? In response to the above response and rhetorical question, I would suggest that in some circumstances, a person with dual citizenship might actually be a more effective president of the United States. John McCain, by the way, was born in the Panama Canal Zone. Many Americans, myself included, were born and educated in the United States, but also have additional education, travel, and work experience abroad. As a global citizen with a broad perspective on the world, I would consider myself and many of my ex-pat and dual citizen colleagues far more competent to run for president than some of our current candidates. The conflicts of interest that an elected official must avoid regarding seats on boards of directors, and affiliations with multi-national corporations, for instance, would be no different from the potential conflict of interest that might need to be addressed by a presidential candidate with dual citizenship. Certainly, there might be potential conflict of interest if the candidate were a citizen in a country that was at war with the United States, but the possibilities for greater reconcilliation and global peace and economic stability might actually be improved through the expanded depth and perspective of those who are multi-lingual, multi-experienced, and internationally educated. Citizenship, today, means much more than the owing of allegiance to a country. Simply because I might have allegiance to the United States, it would not mean that I could not support and even defend a nation with similar principles and objectives. A president, however, when he or she takes an oath of office, does so with full intent to uphold the office itself for the good of the country to which that office belongs.