Could some of these House delegations vote for the independent
candidate if they agreed to do so?
Yes.
Or are the individual members "bound" to the candidate of their own
party?
No. Individual members would usually vote this way and it is commonly assumed that they would, but it isn't required by law. The U.S. Constitution doesn't even recognize the existence of political parties.
Also, since state delegations cast votes as a whole, a majority of the delegation has to support a candidate to cast a vote at all in this contingent House election. If the state delegation can't get a majority of its members to agree on a single candidate then the state forfeits its vote in the contingent House election. And, it is a one state-one vote system in the contingent House election.
The process is set forth in the 12th Amendment to the United States Constitution which states in the pertinent part:
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. But in choosing the President, the votes shall
be taken by states, the representation from each state having one
vote; a quorum for this purpose shall consist of a member or members
from two-thirds of the states, and a majority of all the states shall
be necessary to a choice. And if the House of Representatives shall
not choose a President whenever the right of choice shall devolve upon
them, before the fourth day of March next following, then the
Vice-President shall act as President, as in the case of the death or
other constitutional disability of the President—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; a quorum for the purpose shall consist of two-thirds
of the whole number of Senators, and a majority of the whole number
shall be necessary to a choice. But no person constitutionally
ineligible to the office of President shall be eligible to that of
Vice-President of the United States.
Effectively, the contingent election is a two step process. First, there are fifty intrastate votes over how to cast the state's contingent election vote, and then each of those fifty state decisions are tallied up. To elect a President, at least 34 states need to cast their votes, and a majority of the states voting have to support the same candidate. Otherwise, they get to try again until they elect someone or the VP elected by the Senate takes over.
In theory, a candidate with support only from the eighteen smallest state delegations could be elected President.
Pundits usually assume that delegations with a majority of one of the major political parties will cast that state's vote in favor of the their political party's candidate, and that tied state delegation won't cast votes. But there is room for wheeling and dealing that could produce a different result, especially in a case where there is a third-party spoiler candidate.
Suppose either the Democratic or Republican candidate wanted to
support the third party candidate to deny the other opponent the
election. Could this candidate ask "his" state delegations to vote for
the third party candidate to help him form the needed majority?
Yes. Members of Congress aren't bound to comply with that request, but any candidate (or anyone else for that matter) has every right to ask.
