To your top level question, yes and no. It's complicated, and hopefully the way I explain this should cover your followup questions as well.

So the first thing to understand is that circa 1793 or so there's scholarship to suggest the formation of proto-party voting patterns, which are mostly based on sectionalism as well as a vague lining up behind emerging leaders (although it's worth remembering that Jefferson is technically out of politics when he temporarily retires at the end of 1793 after getting tired of Washington not following his advice and taking Hamilton's instead, especially when the latter annoys him by bloviating for hours in Cabinet meetings.) This division solidifies with the Jay Treaty in 1794, as it is a tremendous win for Northeasterners who trade directly with Britain and a massive loss for Southerners who try to trade to the Caribbean and Westerners who want navigation rights on the Mississippi River. To give a rough idea of why there's so much anger, there's a contemporary comment by someone from one of the latter two groups about how they might as well convert to dugouts for the Caribbean because it's the only thing that would fit under the freight limits; essentially, much of the country is effectively embargoed. The South and West aren't alone; France is furious as they view the treaty as providing favored nation status to Britain, which eventually results in the Quasi-War as the French Navy raids merchant shipping.

The terms of the treaty provoke outrage when they're leaked to the press - keep in mind the Senate's deliberations on everything are secret up through the end of the 3rd Congress, so Federalists view their disclosure as yet another betrayal of good faith conduct - and Jay gets burned in effigy so much that he jokes he could make his way to the Mississippi by that light. As such, this effectively marks the beginning of the first party system with Aaron Burr becoming the de facto first minority leader of the Senate by leading a spirited attempt to block the treaty; it goes through on what we'd now consider a straight party line vote of 20-10.

Concurrently, there's also been a proto-party leadership division on which country to support: France or England. In a bit of a reductionist summary, the proto-Federalists generally tend to be Anglophiles; the proto-Democratic-Republicans Francophiles. There is a terrific old post by /u/jordan42 here which explains what Adams and Jefferson thought of the French Revolution, and it's important to understand part of the the latter's whole throated support of it far longer than his peers stemmed from being present at its inspiring beginnings in 1789 and then returning to the United States before it degenerated. In fact, the name of the political party he founds is borrowed from the Democratic-Republican political philosophy debating clubs modeled after those in France, which Jefferson helps import to the United States.

How much the parties coalesce around their support for France and England is a bit of a chicken-and-egg question - sectionalism, trade, and conflicting views of the powers of the government are intertwined with this - but something to keep in mind is that the later 1798 midterms are a Federalist wave precisely because of substantial fears of France, Napoleon, and the Haitian Revolution potentially inspiring slave revolts in the South. For instance, John Marshall wins election to Congress in Richmond, which until then has been considered what we'd call today a D-R safe seat.

But back to 1796. It is one of the most peculiar elections in American history for a few reasons, one of which is that while the proto-parties were somewhat present in 1792 and 1794, 1796 is the first in which the two parties are specifically organized and voters are well aware that the candidates they are electing on all levels are not just serving as qualified individuals rising to the top (as anticipated at the Convention) but are also representatives of strongly differing ideas of government and foreign policy. It is also one in which Washington still casts an overwhelming shadow; if he runs again, he will win, but given the developments of the last eight years, he will not do so unanimously.

There is another great answer by /u/lord_mayor_of_reddit on Washington's thought process about a third term; he is strongly encouraged to do so by his supporters given that there's no real doubt he'll win, but as early as 1795 has more or less already decided to step down as he's been exhausted with public life - especially with some vicious attacks in the press on him by Jefferson's newspapers that peak in July 1796 when he and Jefferson formally break. But Hamilton in particular wants to delay Washington's retirement announcement for as long as possible for the sake of electoral gain; in some states, elections begin far earlier than November and having Washington still theoretically at the top of the ticket is helpful. Despite being out of the Cabinet, Washington relies on Hamilton to work on the draft of a farewell address he'd begun 4 years earlier - ironically started back then with the help of Madison, who shortly afterwards left and became into the leading political strategist behind the Democratic Republicans - when he'd been talked out of leaving public life before the 1792 election.

Hamilton and Washington trade drafts back and forth over the summer of 1796, but on his end Hamilton's effort is made out of devotion to his mentor and patron - he takes out parts by Washington that present him as an aging patriarch - as well as a political calculus which Hamilton views Washington's announcement of his retirement as a campaign document that will provide one final, massive shot across the bow of those partisan hacks who've opposed the most admired man in America (and, not coincidentally, Hamilton as an individual as well.)

I'll directly quote from Ellis' His Excellency (who calls "sifting through the mound of scholarship [on the Farewell Address] over the last two centuries a bit like joining an archeological dig" given the generational layers of analysis of it) because it's a very succinct summary of the contemporary view held by those reading it when it was released a scant two months before the election:

"In the context of his own time, this was a defense of the Jay Treaty, which repudiated the Franco-American alliance and aligned America’s commercial interests with British markets as well as the protection of the all-powerful British fleet. It was also a rejection of Jefferson’s love affair with the French Revolution as a sentimental attachment, temporarily buoyed by popular opinion but blissfully oblivious to the long-term interests of the American public."

So in short, yes, the Farewell Address was influenced by the Reign of Terror in the context of his overall frustration at the Democratic-Republican support for France, but it's more accurate to describe it as a general broadside at everything the Jeffersonians stood for in opposition to him.