In the year 1800, what shared beliefs and experiences between the Federalists and Republicans allowed the Republicans to keep the nation together?
In the year 1800, you might have thought the US would become divided between the Federalists and the Republicans. This was, after all, the time of the Alien and Sedition Act by which the Federalists tried to prevent the Republicans from exercising their right to political speech. However, I would argue that a shared belief in representative government and the shared experience of the Revolution helped keep the nation together.
You have to remember that, at this point, many of the political leaders in the US (in both parties) had worked together to create the Declaration of Independence and/or the Constitution. They had been together during the struggle for Independence. This meant that they shared a common dedication to the ideas embodied in the Constitution. It also meant that they had been comrades during a very trying time. Both of these things evidently meant enough to the two sides to keep them from really splitting into sides that were openly "at war" with one another.