Electoral College tie

[True Federalist (진정한 연방 주의자)]

A Senate tie with an Electoral College tie is quite possible.

[True Federalist (진정한 연방 주의자)]

I'd say Romney/Biden, but I could see the Democrats in the Senate putting somebody in the VP slot just to screw with Romney for the next 4 years (like Schweitzer or somebody).

No way for someone to come out of left field to win in the Senate.  While the House picks from the top 3, the Senate picks from the top 2.  If it goes to the Senate, even if there is a rogue elector, only Biden on the GOP nominee have even a wildly improbable chance.

[True Federalist (진정한 연방 주의자)]

The Republicans will retain at least 26 state delegations and elect Romney.
The Republicans will also take over the Senate and elect the GOP VP

Wouldn't the VP be elected by the outgoing Senate?

No.  Existing law has the counting of the electoral votes take place in early January as one of the first bits of non-organizational business for the new Congress to attend to.  So officially, they won't know that there no one has a majority until then.  (3 USC 15 calls for it be January 6, but four times since 1984, the outgoing Congress has shifted it to some other early January date for ease of scheduling.)

[True Federalist (진정한 연방 주의자)]

So could they choose Obama?

You forget that under the 12th amendment the Presidential and VP ballots for electors are separate.  The only way for this to be doable,  I guess, is if the Dem electors (at least in states where they can do it by law) switch their Presidential and VP votes (i.e., vote for Obama for VP).

So, what could be done is this. If Dems manage to hold majority Republican state House delegations down to 25 (they don't have to have majority in 25 states themselves - it would be enough to have a few ties) they could block the House from electing the President. If they can, at least, draw in the Senate (w/ Biden providing the casting vote), they should instruct their electors to vote for Obama as VP. Then, the Senate will elect Obama VP, and, due to lack of an elected President, the Vice-President Obama will assume the Presidency under the 12th amendment.

The sitting Vice-President does not get to break that tie.  The twelfth amendment specifies that "a majority of the whole number shall be necessary to a choice".  Even if a Senator got locked in the bathroom and couldn't come to the floor to vote, a 50-49 vote would not be able to elect a Vice-President.  If it goes to the Senate, 51 or more Senators would be needed to elect a Veep, no matter how many Senators vote.

Upon which, President Obama can use the 25the amendment to nominate Biden for VP. The only problem would be to get this through the Senate (once Obama becomes President, Vice-Presidency would be vacant, so there wouldn't be anybody to break the tie.l

That would be a bit of constitutional fun )

Except that wouldn't be the case either.  As per the the 20th amendment, "If a President shall not have been chosen before the time fixed for the beginning of his term, or if the President elect shall have failed to qualify, then the Vice President elect shall act as President until a President shall have qualified".

In other words, under your scenario, Vice President Obama would serve as Acting President until the House reached a decision, and Vice Presidency would not be vacant, so the 25th amendment would not apply Theoretically, that could be because the elections to the next Congress gives one party or the other sufficient control of the House to elect a President.
Now, here's a fun bit of constitutional jujitsu that could happen.

The electoral college is tied, and while the House is unable to reach a decision, the Senate can.  So we get a Vice President who serves as Acting President, pending the election of a President.  But if the Vice President dies in office, then the next in line becomes an Acting President who could use the 25th Amendment to nominate a new Vice President who would take office as Vice President and thus become Acting President.

[True Federalist (진정한 연방 주의자)]

But, is there a likelihood at all that Senate Democrats decide to give Romney his own Vice President in order that the Romney administration will be better able to deal with the tremendous problems facing the nation?

None whatsoever.  Possibly if the Senate ties 50-50 and Romney wins the popular vote a Democrat will break ranks, but otherwise no chance.  Furthermore, unless Romney is incapacitated, it doesn't really matter who the Vice President is, so the idea that any Veep would help or hinder Romney's agenda (or Obama's agenda if he is reelected) is ludicrous.

[True Federalist (진정한 연방 주의자)]

There hasn't been a VP openly hostile to the President in a long, long time. There is a constitutional role for VP that has never been properly exploited. As an opposition VP would not have any real job other than chairing the Senate, he could try to establish leadership there. The Senate is a peculiar institution, so a VP, who knows it well and can build coalitions could do a lot of mischief. Even aside from the fact that in case of a 50-50 tie he'd be the routine casting vote.

True, we haven't had a Veep actively hostile to the President since Mr. Warm Bucket himself, John Nance Garner, was in his second term.  However, unless the filibuster is eliminated, it's hard to see where the Veep's tie-breaking vote would be of much importance.

[True Federalist (진정한 연방 주의자)]

With a different take on this scenario, an interesting statement from the chairman of the MA Republican party, explaining the importance for personal chemistry and the ability of  working well together for the President and Vice President.

Robert Maginn, chairman of the Massachusetts Republican party.

“The second thing is, there’s gotta be good personal chemistry. The president and the vice president have to work together very well. So there’s a personal connection that has to be there, they have to be able to help each other, advise each other.”

http://abcnews.go.com/blogs/politics/2012/05/romneys-vice-presidential-checklist/

That is only because as members of the same party, the Vice-President is expected to serve as a Presidential surrogate to float trial balloons and other politically useful functions.  But governmentally there is no need for the Vice-President to do that, and if the Vice President is of another party, then the situation would be the same as in those states where it is possible for the Lt. Governor and Governor to be of opposing parties.  That is, the Vice President would be the leader of the opposition.