2) Can a president who never served as vice-president run for vice-president after their terms as president have expired?
From the XIIth Amendment:
"But no person constitutionally ineligible to the office of President shall be eligible to that of Vice-President of the United States."
From the XXIInd Amendment:
"No person shall be elected to the office of the President more than twice, and no person who has held the office of President, or acted as President, for more than two years of a term to which some other person was elected President shall be elected to the office of the President more than once"
You'd have to cook up some very convoluted logic to argue that a person who is ineligible to be elected to the office is somehow still eligible to the office. It certainly woud go against the clear intent of the XXIInd.
However, I think it would be possible for such a person to become acting President under the XXth Amendment in the case of a failue of either a President or Vice President to be elected, who would serve until one or the other office were to be filled.
But someone who is term limited is not constitutionally ineligible to be President. Clearly, if Bill Clinton was Secretary of State and the requisite people died, he'd become President. The 22nd amendment is merely a limitation on who can be elected President.
Of course none of that would stop the 5 partisan activist conservative Republicans on SCOTUS from vetoing Bill Clinton being elected VP.
I'm far from sure, because there have been Secretaries of State, Kissinger and Albright, that were not eligible to be president; Kissinger was was three spots away at two points. There usually are ineligible people in the line of succession.
I'd make the same argument about G. W. Bush, however. In theory, in 2009, he could be elected Speaker; there isn't a requirement that the Speaker be a member. I would question if he could become President in that circumstance.
This text from Article II states that a foreign-born citizen couldn't be president at all. Now, were George W. Bush the speaker, and he ascended to the presidency, he could constitutionally serve, as the 22nd amendment merely states that a two term president could not be
elected for a third.
There's a different between being ineligible for the presidency and ineligible for election.
