Speaking of which, if he were convicted and then elected President, one could make a reasonable argument that he could then pardon himself.
So technically speaking, a murderer could be elected president, but not a legal immigrant.
Well, that depends, particularly for Snowden. "Under Section 3 of the Fourteenth Amendment, no person who swore an oath to support the Constitution, and later rebelled against the United States, can become president. However, this disqualification can be lifted by a two-thirds vote of each house of Congress."
Federally I am pretty sure the answer yes, everything else would depend on the states I guess. In fact you can even be in prison I am pretty sure.
Bush and Obama both got elected for second terms...