I’m getting very confused about how the change in Vice President appointments were made. So, I get that in 1796 Jefferson lost, and he then became VP. He then ran in 1800 against Adams and won. Why didn’t Adams become VP? Did he not want to? I know the election was nasty, and he didn’t even turn up for the inauguration. The 12th amendment in 1804 changed the system, but that was after the election of 1800 (obviously haha). Then what I don’t get is why were there running mates? Even if something changed before the 1800 election I don’t know about, that process (loser to VP) still was enacted in 1796. Why did Adams and Jefferson have a running mate for that election, when the loser of the election would be VP. In case a crisis happened? Also, when did the change happen that president and VP were elected together. I think that became more of a thing later in the 1800, but when were they first “stuck” together?
Lastly, while I’m interested in my next question, I’m far more interested in the above information. Why (maybe also how) was John C Calhoun the running mate for John Quincy Adams as well as Jackson in the same year?
Hopefully I made sense!
Prior to the passage of the 12th Amendment, the Constitution specified that whoever got the second most votes among the electors would become the Vice President. The quirk of the system came about because each elector gets two votes, and the rules specify that one of those votes must go to someone from outside the elector's home State. The thinking at the time was that State and regional favorites were likely to emerge, and it was quite possible, or likely even, that the "electoral college" (a term that is used only by convention and does not appear anywhere in the Constitution) would act more as a "nominating body" to determine which candidates had the broadest support, and cut the field down to a manageable number, hence Article II's requirement that the House choose among the leading five vote recipients if none received a majority (https://constitutioncenter.org/interactive-constitution/article/article-ii ).
What apparently few of the framers accurately anticipated was the relatively early formation of party politics. The first few Presidential elections were largely uncontested because of the stature of Washington, but by 1796 it was quite clear that "factional" (or Party) lines were forming around Adams/Hamilton on the one hand, and Jefferson/Madison on the other. But this organization was relatively poor in 1796, and there certainly were not organized "tickets" for the parties--henceforth referred to as Federalists (Adams) and Republicans (Jefferson)--in the style that would be standard at a later date. Hence Federalists just made sure they had enough votes to elect Adams, and split their second votes among a variety of candidates, leaving Jefferson with enough votes to take the Vice Presidency. Nonetheless, there are definitely signs of strategic thinking in the election, as seven States opted to select their electors via their State legislatures, ensuring that whichever faction controlled the legislature would control the State's electoral votes.
By 1800 the situation had evolved enormously, and parties were thinking on clearly strategic lines--at this point, most States assigned their electors based on control of the State legislatures, allowing parties to tightly control their electors. Notably this was not the case in Maryland, North Carolina, nor Kentucky, of which the first two produced a slate of divided electors (as did Pennsylvania, but this was because of divided rule in the legislature, not district-based selection of electors). Under the rules of the system, if a party is to behave optimally, they should select two candidates from two different States, and ensure that ALL of their voters cast one vote for whom the Party wishes to be President, and all but one vote for the person whom they wish to be Vice President, so that the secondary votes are not wasted to potentially allow a rival to assume the Vice Presidency, as had happened in 1796. This is where the idea of a "running mate" comes from. The problem in 1800 arose because for reasons that are still somewhat disputed, every Republican elector cast one vote for Jefferson, and one vote for Burr. Because the election was tied, the House had to vote only among these two candidates, not the leading five. Adams at this point was eliminated--either Jefferson or Burr would be President, and the other would be the Vice President, depending on how the House vote went (bolding this section because it directly pertains to one of your questions). After much debate and politicking, the House settled on Jefferson (though, side note here, the oft-told story about Hamilton influencing Federalists to switch their votes from Burr to Jefferson does not have a lot of evidence to back it up--certainly we know Hamilton tried to influence them, though whether he actually did is a subject of contention).
The 12th Amendment simplified this situation considerably, and stipulated that when the electors cast their votes, one would explicitly be for President, and the other for Vice President (https://constitutioncenter.org/interactive-constitution/amendment/amendment-xii ). This more or less formalized the "running mate" paradigm, though it's worth noting that these are technically separate votes, and it would be theoretically possible for a modern elector to cast a Presidential vote for Joe Biden, and a Vice Presidential vote for Mike Pence--Party rules and discipline of members largely prevents this, although occasionally shenanigans and protest votes do occur, as recently as 2016 (but we will not get into that, keeping to the 20-year rule).
Further reading and sources:
Hopefully that answers a bit. I will have to either check my books or let someone else address the subject of Calhoun in 1824, though again I will note that since the vote for Vice President was entirely separate than the vote for President, and Party organization was again weak in 1824 (becoming strong again in 1828), it was feasible for him to seek the Vice Presidency outside formal party selection--though I'm a little short on details right this second.