"President elect" means the candidate obtained a majority of legitimate electoral votes, not that this person is qualified.
The form of objection is unstated, beyond having to be in writing and clearly, concisely, and without argument, stating the grounds for objection. See, perhaps, for a parallel, the argument in Congress surrounding the seating of Smith (1789) on account of his not being a citizen for the required 7 years.
Given the objection, or question, Congress is required to vote on the objection. There is a time limit for debate, 2 hours! (3 USC 17).
3 USC 19 states the possibility of vacancy in the office of president, by reason of failure to qualify. If neither the president elect nor VP elect qualify, the Speaker (or whoever down the line of succession is qualified) becomes president until there is a qualified president, which may take as long as the 4 year term.
-- If contested on grounds of Constitutional ineligibility it would require a judicial decision because each state is responsible for its own election and it would be members of Congress pitted against a state SOS claiming that the candidate was eligible ... --
I think the issue of eligibility would run against the political party, not the state. From the state's perspective, the voters have chosen an elector, not a president - and it is up to the elector to vote for a qualified candidate.
-- If members of Congress had contested, say, California's electoral votes because the Keyes case was still pending, how should the rest of the story go? --
Congress has a free hand here. If Congress has a question about the qualifications of the person obtaining a majority of the electoral votes, it needs to settle that issue, publicly.
I think Congress is utterly corrupt and derelict. IT can see the party candidates coming from months away. It saw fit to paper-over McCain's qualification in advance. Why not a similar papering over for born dual citizen Obama?
I think Congress-critters would claim that they couldn’t legally do anything until the electoral votes were opened because until then it would just be speculation. But that sure didn’t stop them from dealing with McCain speculatively, did it? Jerks.
I suppose it’s ultimately the political parties who assign the electors who do the actual voting that counts.
Which probably means the legal buck stops right there at Nancy Pelosi and Alice Travers Germond. They gave the signal for their electors to vote for Obama because they promised he was their duly nominated candidate (i.e. fulfilling all the requirements of the DNC, which includes Constitutional eligibility). In Hawaii they went even farther and personally testified outright that they knew him to be Constitutionally eligible. Liars.
The more I think about it, Congress probably could have contested the electoral vote. Those electors could have voted for Mickey Mouse, and would Congress have to certify Mickey Mouse as the electoral winner?
But I don’t think Congress could by itself determine whether Obama is a “natural born US citizen” - unless to rule him out as one because of his age, residency, or birthplace. The dual citizenship issue probably involves interpretation of the 14th Amendment and so would require a judicial ruling.
And I think the 20th Amendment provides for a situation where Congress certifies the winner but the winner still doesn’t qualify - which just about has to mean that somebody is able to bring a case and find that he failed to qualify.
The trouble is that our courts refused to hear the cases.
What a fine mess.