Hillary Clinton/Barack Obama Ticket To Replace Joe Biden?
That cannot happen because of the 12th. Obama is not eligible to run as either president or vice president.
Can someone appoint him as VP (subject to approval by congress) if the elected VP should resign or die? Yes.
But even so, he cannot stand for election for VP when the president who appointed him VP wants to run for reelection.
A lot of these amendments have not really been challenged in court as far as I know.
As I recall, Theodore run for president again in 1912, after serving 3 years of McKinley's presidency then winning in 1904 to serve another 4 years, making 7 years altogether, then taking 4 years off to do other stuff like hunting in Africa. But then that was long before the 22nd which became law in 1951 if I recall correctly.
He can't be elected vice-president but he can be appointed to it? By that logic then someone who is 25 years old can't be elected vice-president but could be appointed vice-president.
What is "running"? The Constitution doesn't say anything about "running".