You are correct to say that the states are legally free to require proof. They will not do on their own volition in the present political environment, however. As long as Obama's political fate might be in the balance, we've seen cold feet everywhere, even among Republicans, when the issue comes up in the state legislatures. Nah, no one wants to be branded a "birther", because the next things you'll be called are "racist," "demented," "paranoid," or any combination thereof. Some Republicans might also be thinking that such legislation might negatively impact Marco Rubio or Bobby Jindal at some point in the future.
I don't think it would even take an amendment. It could be done by law.
It could be argued either way as to whether the "necessary and proper" clause is applicable to enforcement of the constitutional eligibility requirements for the presidency and vice-presidency. For all their wisdom, the Founding Fathers laid out no specific means of enforcement of those requirements, perhaps relying solely on the good faith of prospective candidates to abide by such requirements. They did not explicitly delegate such an enforcement power to Congress.
Happens in EVERY election. Any state AG can throw Obama off the ballot and order him to show how he is constitutionally qualified. BTW, same is true for any candidate.
No guts. And no AG apparently wants the glory.