@gourimoko, since you're the one with a huge amount of political experience... serious question here...
What would happen if one of the presidential nominees dies a few days before Election Day?
Depends on the individual states and their rules. In the majority of states, the Secretary of State would remove the deceased candidate and allow the local state party delegates to appoint a new candidate. Thus, it would be unfortunate for that party, but, the election would assuredly continue.
Does Congress/SCOTUS postpone the election until that person's party can nominate a replacement and hold a few debates with the other party's candidate?
Definitely not. Under no circumstances is this allowed.
If the candidate dies before the election, the election continues on schedule and the individual states would carve a path for the party to nominate a new candidate.
If the candidate dies after the election, then the electors of the state determine the new
candidate under their own discretion; this person is not a President-Elect since the electors would no longer be bound by their oath to uphold the democratic will of their state. State electors are often party delegates anyway; so again, it'd fall to the local party. This doesn't mean they'll reach a consensus of 270 electors though! Which means the election's outcome is not necessarily assured for a given party. This could result in Congress selecting the President; however, the most likely result would be a party caucus/convention of delegates and super-delegates to nominate a President.
If the candidate dies after the electors have convened, but before being inaugurated, the Congress selects the President. There would be no special election.
If the President dies after being inaugurated, the Vice President becomes President.
Under no circumstances can there be a revote, a delayed vote, or a special election in our system of government.
This is the reason Bush was essentially installed as President as there was no way to invalidate the Florida election. (I voted for GWB in that election, FWIW)
Does Congress/SCOTUS even have the authority to do that?
None.
Congress can appoint the President, but only under very particular circumstances.
Or would that party simply be S.O.L. for that term's election?
This.
They'd likely select an extremely popular person and simultaneously select the current sitting President (if of the same party and popular) as the VP nominee. So, if Clinton died, they'd likely select a very popular Democratic candidate to lead the ticket, and use Obama as the VP nominee to get name recognition on the ticket.
You could envision a Biden/Obama ticket. That would be the most logical choice if there was no clear runner up during the primary season. As in, if Clinton wins Iowa, New Hampshire, and South Carolina, the primary season is over. If she were to die, there would be no mandate for a second place finisher. So the logical choice would be Biden. His name recognition isn't that high, so picking Obama as a running mate would make sense.
How would the country react to that?
Depends... If the candidate was popular, then there could be an outcry towards that party. Look at LBJ in 1964.
Obviously it's never happened before, but I'm curious to hear what you think would occur.
It's actually a question that's come up many times; we have provisions for any such occurrence.
The real problem is if the electoral college is split 269/269, the popular vote goes to one candidate, and the House is controlled by the opposing party. That is a Constitutional crisis in the making.