I dispute the core assumption, that Fremont was close to winning. Fremont was more liberal than Lincoln and directly stated his opposition to slavery, rather than Lincoln's finely calculated stance in 1860. However, electoral WIs are always fun, so let's do the numbers:
First, there's no way Fremont can carry states in the South. The only states he can possibly flip are, Pennsylvania (27 EV), New Jersey (7 EV), Indiana (13 EV), California (4 EV) and Illinois (11 EV) -- a total potential swing of 62; Fremont needs 35 more votes than he won OTL. If could sweep all of these, he wins (you needed 149 votes for a majority), but with Buchanan running there's no way he loses Pennsylvania. He was a native - son candidate to begin with.
The major variable is to tinker with Millard Fillmore's Whig-American Party, since it made the contest a three way race. (See
here for popular votes by state). Note that I'm assuming that if Fillmore evaporates, Maryland votes for Buchanan, since Fremont got fewer than 1,000 votes in slave states OTL.
In Indiana, even if Fremont got
all of Fillmore's votes, Buchanan wins 50.7% vs. 49.6%. Considering that the Governor of Indiana (both the incumbent and the candidate who won) was a Democrat, I'd imagine there might be an even bigger margin for Buchanan in such a close race. Amazing how you can find votes down in the southern coutnies.
In California, Fremont even as a native - son only got 18% of the vote, so he can't win there either.
Assuming a best case scenario (simply adding Fremont and Filmore together), Fremont might win in New Jersey, but that's a very tenuous assumption.
Illinois probably has the best chance of swining for Fremont, since Filmore got 15.7% with Buchanan at 44.1% and Fremont at 40.1%. Without too much alteration, you could probably detach enough Buchanan voters to get Fremont a plurality.
The tally is this:
States Fremont can't win: California, Pennsylvania, Indiana = 44 EV.
States Fremont might win (but didn't): Illinois, New Jersey = 18 EV.
Hence, if Buchanan performs similarly, Fremont and the Republicans can't win. Indeed, positing the non-existence of the American Party probably isn't valid since it requires the Republicans to instantly (in 2 years) build a very wide coalition. The Republicans might have sought to include a broad platform -- support internal improvement, a high tarrif and the like -- and if so, they can probably seriously limit the appeal of the American Party and perhaps run a joint ticket. However, I think this is unlikely since the Americans were the much stronger party in 1856, controlling the Speakership in the House of Representatives.
For Fremont to have a prayer, Buchanan and the Democrats need to have horrible luck. I'm thinking Buchanan needs to die in mid-September, forcing the Democrats to call an emergency convention to chose a new nominee. If they actually can call such a meeting (rather than simply fragmenting), then it's likely to be a show-down between Pierce and Douglass. OTL competition between these two allowed Buchanan to emerge as the compromise candidate. Breckinridge isn't well-suited to be a compromise candidate. Pierce or Douglas prevailing might prompt factions of the Party to bolt, but that probably favors Fillmore and the American Party. The only two compromise candidates I can think of are those sought by the Constitutional Union Party in 1860: John Bell and Sam Houston. Though you might have others: Jefferson Davis might be fair game, ironically. My Lone Star Blinders have me thinking that Houston might have a decent shot at uniting the Democrats, but I think he'll have few supporters; probably you need someone with a bigger base. Douglas would probably have to fall on his sword.
From here there are a lot of options. If the Democrats are only weakened in the South, or if substantial numbers bolt to the Americans, then there's a real possiblity that no candidate has a majority in the Electoral College, leading to a House vote to decide the Presidency. Douglas probably has the edge here, but it's hard to tell and it's similarly hard to tell who he'll favor. The hard thing is if 1860 really does happen early and the split in the Democrats weakens them in the North and favors the Republicans. This likely depends / requires the Republicans to be more than just a free soil party, but as per the above, it may be possible to acheive a majority -- barely.
As to the second question: how does Fremont respond to the Civil War? He probably will want to make it into a crusade to liberate slaves from traitors, judging on his actions as a general. He may also have a greater understanding of military command and find more able generals more quickly; or he doesn't need to because Winston Scott has enough energy to serve as General in Chief. Bringing slavery into the war very early complicates things in the North: first, it forces the hands of Kentucky and Missouri. These states may outright secede -- a big blow to the Union war effort. Second, while you may get a tide of abolitionist recruits, you're likely to cause a lot of dissent in the North if Fremont starts talking about "Negro equality." Third, there's not really a clear way for Fremont to take action that doesn't galvinize the South or trample on Federal legal structures (or the Constitution).
Also, Pierce probably reacts very differently to a Southern secession than Buchanan did. I can't see him sitting on his hands, but his loyalties will be very conflicted, being President and a Southerner at the same time.
The differing international scene may be particularly telling: if the war errupts in 1856 in a similar time frame, then the Indian Mutiny will be in full swing. This may distract Palmerston but it also may make it more difficult for the textile industry to find replacement cotton. You also have more chaos in British politics, with Derby's minstry in 1858.
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One more thing I forgot to mention. The American Party had signifcant support in New York, as well; its support there was sufficient to help edge the state to Fremont (though not without some genuine support Up-state). If the Americans do fair poorly, it may be hard for Fremont to count on holding his gains here.