1) Can a former president who was elected to one term but lost their re-election bid, run in another election for their second term?
2) Can a president who never served as vice-president run for vice-president after their terms as president have expired?
1. Yes, the prohibition is only against serving or running for a third term, not for running for a second term multiple times. So George HW Bush or Jimmy Carter, for example, certainly could run for President again legally. George W Bush and Bill Clinton, of course, could not run again.
2. Yes, as long as the ex-President is still eligible to be President. If he served or was elected to two terms as President (with a term served being defined as at least half of a four year term), he cannot serve as VP; the VP must meet the same requirements as the President. Whether the ex-President ever served as VP in the past is irrelevant.
Bill Clinton and George W Bush could not be VP, but George HW Bush or Jimmy Carter could.
