The original Republicans, the Federalists, the National Republican Party, the Democratic-Republican Party, Torries, Whigs, Southern and Northern Democrats, there's easily a dozen political parties that have put candidates into the Presidency but no longer exist.
My question has 3 main parts:
To the general public what did the final years of these parties look like?
What were their final candidates like?
What happened in the years between the death of one party and a return to the equilibrium of a 2-party system?
Edit: In the interest of answers feel free to cherry pick which particular question you'd like to answer
The last time a major political party died was the Whigs in the lead up to the Civil War. The Whig Party broke apart on the question of slavery. Northern factions became more anti-slavery, while Southern factions refused to abandon slavery. The Party could not contain these contradictory ideas, so it lost support and quickly found its members deserting the Whig Party for alternatives.
As the former Whigs began to abandon their party, new political parties appeared to take them in. Those parties included: the Free Soil Party, the American Party (sometimes known as the 'know-nothing' party) and the Republican Party. By the election of 1856, the Whigs were gone.
Interestingly enough, the Democratic Party also split on the issue of slavery in 1860, with Northern and Southern factions emerging to nominate their own candidates. However, the Democrats were able to recover after the Civil War and continue to be a major party to this day (of course).
The other major parties that died (The Federalists, Democratic-Republicans, National Republicans kinda) weren't really political parties in the sense that we understand them. They were more alliances of elites competing against each other, as opposed to mass mobilizing voters. The Federalists died largely as a result of the total victory of the Democratic-Republicans and the Democratic-Republicans also died largely as a result of their victory, leading to the somewhat party-less period known as the 'Era of Good Feelings.'
All the other parties you mention were minor parties that were either formed as result of a brief split from the major parties (Southern Democrats) or as a the result of a single influential man creating the party as a platform to run on (the Progressive Party).
In a sense the only true major political party that has died was the Whig Party.
So now comes the real question, why has there not been another party collapse in the 150 or so years after the Civil War? Why have we stuck to the Democrat/Republican divide, even as those parties have changed radically both in supporters and in issues?
The answer is that absent an issue so divisive as that it literally led to civil war, parties are pretty damn durable. Every time a major challenger to the two parties has emerged (such as the Progressive Party in 1912), one or both of the two parties have adjusted themselves and their issues to try to be welcoming to those voters and issues. Thus the Democratic Party moves from being a small government party in the 19th century, to being a progressive party in the early 20th to being the party of the New Deal in the mid-20th century.
In America's two party system, which is reinforced by our first-past-the-post system of elections, parties should be viewed less as solid ideological actors and more as alliances of disparate interests that come together in order to seek political advantage. Thus you have labor and environmentalists largely in the same party, not because those two views are immediately reconcilable, but because it is an advantageous political alliance. When those alliances break down, groups may switch from one party to another (something called 'realignment'). Thus the two parties survive, even as supporters and issues may change.
This is quickly veering into the realm of a political science discussion, so I'll just end here with a few quick answers to your questions.
Sources: https://www.amazon.com/Battle-Cry-Freedom-Civil-War/dp/019516895X https://www.amazon.com/John-Quincy-Adams-American-Visionary/dp/0061915416/ref=tmm_hrd_swatch_0?_encoding=UTF8&qid=&sr= https://www.amazon.com/Bully-Pulpit-Theodore-Roosevelt-Journalism-ebook/dp/B00BAWHPX2/ref=sr_1_1?s=books&ie=UTF8&qid=1468985270&sr=1-1&keywords=bully+pulpit+doris+kearns+goodwin#nav-subnav https://www.amazon.com/Presidential-Campaigns-George-Washington-Bush/dp/0195167163
Incidently, has there ever been the rise of a party in the fashion of the Liberal Democrats in the UK that shook down the US bipartisan system?