Can Pres. Clinton be Kerry's VP?

[Emsworth]

They can choose a 3rd place candidate can't they? How does that work, are they allowed to choose someone who didn't recieve EVs?

No, the Senate must choose between the two highest vote-getters. The Twelfth Amendment states:

The person having the greatest number of votes as Vice-President, shall be the Vice-President, if such number be a majority of the whole number of Electors appointed, and if no person have a majority, then from the two highest numbers on the list, the Senate shall choose the Vice-President...

The Senate cannot choose a person who has received no electoral votes, either.

[Emsworth]

Hm, OK, what about president. I might be mixing them up. Is the House allowed to choose a 3rd place in EVs for president?

Yes. Again from the Twelfth Amendment:

The person having the greatest Number of votes for President, shall be the President, if such number be a majority of the whole number of Electors appointed; and if no person have such majority, then from the persons having the highest numbers not exceeding three on the list of those voted for as President, the House of Representatives shall choose immediately, by ballot, the President.

[Emsworth]

Ah, that's it then. So this basically means that in the, highly unlikely, event of 3 candidates of which one belongs to a party controlling both the House and the Senate but still ends up in 3rd place in the EC could get the presidential spot but not the VP one?

Remember that the third place candidate may not necessarily win if his party controls the House of Representatives. When the House votes, each state has one vote, and it is entirely likely that the party controlling the House is not the one controlling a majority of states - especially given that three major candidates, and perhaps therefore parties, are apparent in such a hypothetical.

[Emsworth]

lets say during the middle of Clinton's second term he died. Gore becomes president for the remaining two years of Clinton's term. Gore runs for the first time and wins in 2000. So Gore is president for 6 years but only runs and wins 1 term of 4 years. Would he be able to run again in 2004  even if it meant he would end up being president for a total of 10 years?

Amdt XXII: "no person who has held the office of President, or acted as President, for more than two years of a term to which some other person was elected President shall be elected to the office of the President more than once."