If there is a Constitutional basis for it. There is plenty in the Constitution detailing how to handle a vacancy in the presidency, but nothing in the Constitution allowing a do-over on elections.
Right. There would not be any new election because the Constitution does not provide for that. It provides for the House of Representatives to pick a President.
If the Court declared Obama ineligible, there would be no vacancy in the presidency, because the VP elect (Biden) would become the Acting President.
There would be no "do-over election", because the XXth applies without requiring another election.
But none of that will happen, because the USSC will not allow the case to be heard.