Most likely IMHO? Obama would resign in favor of Biden regardless of whether impeachment was threatened.
If people think that the Republicans are being obstructionist now, just imagine how bad it would be if they were realistically able to claim that he has no authority do do anything but empty wastebaskets. I expect there would even be some moderate Democrats who wouldn’t want to give the impression they were supporting fraud upon the US.
Biden would be seriously hampered in anything he tried to accomplish, but Obama would be completely hamstrung.
Very, very likely. If you remember when Nixon was embroiled in Watergate, everything else pretty much went on the back burner with respect to governing the country. He resigned when he did partly to avoid stretching out the agony any longer, even though he could have continued the legal battle (which he certainly would have lost) to the bitter end.
I doubt Obama would want to taint his place in history any more than was necessary.
Pretty much none. AFAIK these are the only ways to become the President of the United States:
[ul]
[li]Being elected by a majority of the electors. (Nope, that was Obama.)[/li][li]Being elected by a majority of the state delegations in the House of Representatives after there was no majority vote by the electors. (Nope, Obama got a majority, so they didn’t even vote.)[/li][li]Being elected Vice President by a majority of the electors and filling a vacancy in the presidency. (Nope, that was Biden.)[/li][li]Being elected Vice President by a majority of the Senate after there was no majority vote by the electors, and then filling a vacancy in the presidency. (Nope, the majority of the electors voted for Biden, and Romney wasn’t one of the top two vote-getters for VP anyways. That was Paul Ryan.)[/li][li]Being appointed Vice President and confirmed by the Senate and the House of Representatives, and then filling a vacancy in the presidency. (This at least isn’t impossible, just vastly improbable.)[/li][/ul]
Other people can become the acting president, to exercise the powers of the office until there is a president*, but sadly for Mitt Romney, he isn’t any of those people either.
*It’s been theorized that Congress could call for an early election if there wasn’t a president. When it seemed possible that Nixon would be forced out during the vacancy in the vice presidency, that possibility was floated.