What were the major areas of disagreement and debate about the nature and purpose of democracy in the United States in the late 1780s and 1790s, and to what extent (and in what ways) were these different ideas incorporated into the “revolutionary settlement” that emerged in the wake of the election of 1800?

The birth of American democracy is an interesting and complicated topic. Though there are specific dates and key events we can point to—like the signing of the Declaration of Independence on July 4, 1776, or April 30, 1789, when the nation's first president assumed office—the meaning of democracy, and the ways in which it would function in the United States, was a subject of great debate among founding fathers, statesmen, and intellectuals.
Let's go over a few of these major areas of disagreement. One central question was what is the purpose of democracy? Is the objective for individuals to have the greatest possible freedom, or is the goal for the community to be strong and unified? In other words, is democracy about the people, or is it about the larger system they live in? 
Another early area of discussion centered around the concept of checks and balances within the three branches of government. John Adams, a federalist, strongly believed in the need for checks and balances, pointing out that one branch could wield a disproportionate amount of power if the other two didn't have the power to balance it. Not everyone agreed with Adams's idea, and many were uncertain about his belief that each of the thirteen states should remain sovereign. In Adams's view, each state should be its own republic, and each should have its own constitution. 
Unsurprisingly, another key conflict had to do with money. Some nationalists (including, notably, Alexander Hamilton) believed that the national government needed to tax more heavily, while others disagreed. By the late 1780s, American farmers faced an economic crisis: they had high taxes to pay and debt that they couldn't pay back. Meanwhile, an expanding Army and Navy were expensive for taxpayers. That was controversial, too.
Also contentious was the subject of representation. Assuming that each of the thirteen states remained independent, how would each state be represented in a national government? In the early days of the democracy, between 1777 and 1781 (when the Articles of Confederation were finally ratified), there were significant delays in deciding how this would be arranged. In fact, it was a slow process trying to get any law passed under the Articles: nine out of thirteen states had to approve a law in order for it to pass. Each of the thirteen states were advocating for their own best interests, and it took years to come to an agreement. 
The list goes on. But let's skip ahead to the election of 1800 and the legislation that followed it. The election had Thomas Jefferson going up against John Adams. By this point, Adams was a clear symbol of strong, centralized federal power: he wanted to further expand the Army and Navy and introduce new taxes. Jefferson, in contrast, stood for power to the people: he wanted to cut back the nation's authority and give more power to state governments.
Jefferson won (which you'll remember pretty clearly if you've seen Hamilton).
So how did the hot topics and controversies leading up to the election of 1800 show up in the writing of the legislation that followed it? Jefferson focused on uniting the two sides, aiming to establish the United States as the "strongest government on earth." He helped to reign in the national government's authority, giving more power to individuals and to state representatives (going back to that fundamental question about what the purpose of democracy is). He cut taxes and shrank the Navy: it was peacetime, and he didn't see the need for the extra expense. 
https://www.smithsonianmag.com/history/thomas-jefferson-aaron-burr-and-the-election-of-1800-131082359/