Part One: The Constitution is very blurry when it comes to whether a President who has served two terms (and is thus limited from another term due to the 22nd Amendment) can run as Vice President. There are different interpretations about it, and it’s never been tested.
My interpretation is no — a term limited President cannot be elected Vice President. The 12th Amendment states that “no person constitutionally ineligible to the office of President shall be eligible to that of Vice-President of the United States.” Since we’re using Clinton as an example, I think that the 12th Amendment would indicate that Clinton couldn’t run for Vice President because — due to the fact that he’s already been elected to two terms as President — he is Constitutionally ineligible to seek the Presidency again.
By the way, President Clinton himself interprets the Constitution that way and doesn’t believe he could be Vice President.
Part Two: Yes, there was briefly some consideration at the 1980 Republican National Convention to naming Ford as Reagan’s running mate. Reagan had approached Ford about the VP slot privately before he even locked up the nomination, and Ford declined it, but at the Convention, it became public knowledge, so although he was annoyed about the proposition leaking, Ford seriously considered accepting.
The idea didn’t last long. When Reagan and his aides had first mentioned it to Ford, it was with a promise that Ford would have significant, extraordinary powers as Vice President — almost a co-Presidency. At the Convention, Reagan’s people and Ford’s people quickly realized that they couldn’t agree on what Ford’s role would be, so things rapidly fell apart. It was probably for the best anyway because Ford and Reagan didn’t really like each other. There had been some bad blood between them since Reagan challenged the incumbent President Ford for the 1976 Republican nomination.