I know that early Modern English, being a Germanic language, meant that the first letter of nouns were capitalized; this can be seen in documents such as Paradise Lost (1667) by John Milton and the Declaration of Independence of the United States (1776); however, it is not seen (from what I can tell) in Johnathan Swift's A Modest Proposal (1729); by the time Edgar Allan Poe wrote The Fall of the House of Usher (1839) the practice seems to have completely fallen out of favor with most authors and poets of the time.
When exactly did people stop writing like this? Was Swift one of the first, or is that simply a modernization of his writings so as to not make it too strange in appearance to the average reader of today?
There are no concrete written records as to exactly when and why it happened. There are strong theories, but it is still mostly speculation. Here are some you might enjoy:
These are the main theories, but a highly profilic writer (esp. one who despised the Germans and appraised French literature) could have advocated a more French spelling and influenced literature.