A couple of questions.

[Gabu]

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?

See Grover Cleveland.  He's historical proof that you can.

2) Can a president who never served as vice-president run for vice-president after their terms as president have expired?

I'm not sure.  I could have sworn that one must be eligible for the presidency to be eligible for the vice-presidency, which, if true, would mean the answer is no, but I don't fully recall.

[Gabu]

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.

Right, I suppose "elected" is the key word.

[Gabu]

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.

But the twenty-second amendment limiting a president to two terms says that someone cannot be elected more than twice.  It says nothing about becoming president through the line of succession.