Does an American nation exist?

Yes, America as a nation still exists. The elements of nationalism that contribute to this have been forged by occurrences in history, specifically our independence from England, the addition of land from France in Thomas Jefferson's Louisiana Purchase, defining of our national borders supported by the defining of the border between Texas and Mexico as the Rio Grande river, and other things such as the California Gold Rush of 1849. The Civil War divided us and yet showed our strength as a nation. The addition of Hawaii and Alaska expanded our shipping industries and allowed for military bases off our continental borders. There is simply too much to write about that could be covered in your question, but these are some good places to begin your research.

No, it has recently been discovered that the Treaty of Paris of 1783 was fraudulent, and the United States of America hasn't really existed at all.

Upon hearing the news, the government of Germany petitioned the United Nations to have its defeats in World War I and World War II expunged from the historical record, and demanded the full restoration of the German Empire as it existed in 1913.