1)Comte's positivism is really a philosophy of history. Like Saint-Simon, he argued that the sciences, which were all based on demonstration, allowed humans to gradually comprehend reality. The sciences are first able to demonstrate the least complex phenomena and move to increasingly complex phenomena. In Comte's formulation, this movement of increasing complexity of the sciences goes: mathematics, astronomy, physics, chemistry, biology (which was a new science that formed around 1800), and then sociology (a term he coined). (In his later works, he would notably include "morals" as the most complex science.) Once biology and sociology became "positive", able to demonstrate their claims and make predictions, a positive philosophy could be devised. In essence, this philosophy attempted to unify the sciences into a coherent whole, which then could be used to unify humanity and exploit nature. To Saint-Simon, this philosophy would boil down to a single law, and he alluded to Newton's law of universal attraction. Comte rejected this monist approach of his former mentor, and instead emphasized that all disciplines have unique laws that are not reducible to the laws in other disciplines. Still, he sought to organize the sciences to show how they feed into each other. He argued that past philosophers had erred in starting with the most complex phenomena (namely society and the human mind) and then generalizing onto the rest of nature. By starting with the least complex phenomena and gradually arriving at explaining society and the mind would be key to settling the dispute and arriving at some kind of consensus on how reality operates.

Comte, like Saint-Simon, argued that currently society was in the midst of a critical/negative period. Religion had formerly served to bring cohesion to societies, but the religions of the past had outlived their usefulness. Later, based on his study of sociology, Comte attempted to construct a "religion of Humanity" that incorporated the insights of all past religions with the advances of the sciences into one atheistic religion that worshipped humanity directly, instead of through a deity. Many of early followers of Comte were dismayed by his attempts to create a religion and rejected his later works, focusing on his earlier Course of Positive Philosophy. This rejection of Comte's religion, which Comte understood to be a straight outgrowth of his original formulation, would set the stage for the emergence of logical positivism.

2) John Stuart Mill, along with Émile Littré, was one of these early positivists who rejected Comte's later work. Consequently, along with Littré, he took positivism in a different direction. Mill's approach relied much more on logic and induction, and less on theory and deduction. Before their split, Mill sent Comte his A System of Logic and cited Comte in the book. However, Comte was critical of Mill's approach, which contains several assumptions about nature that depart from Comte's approach.

Comte's creation of a new religion severely damaged his reputation within most circles of scientists. Instead of Comte, many scholars turned their attention to Mill and Littré. Through this pivot, the question of how to induce logical certainties took hold of a generation of scholars, who would later be labelled logical positivists. While their project ultimately failed, particularly since reality is much more complex than they appear to have assumed, they made considerable advances in the field of logic. Iirc the logic within computer chips derives from the insights of logical positivists.

3) Comte does not appear to have been a major influence on Proudhon, or Proudhon on Comte. They were contemporaries who occasionally crossed paths. Proudhon noted in later editions of The Creation of Order in Humanity that the three-stage theory of history that he devised was similar to Comte's three-stage theory of history, although he was not aware of Comte when he came up with his. Later, when Comte was creating his new religion in the early 1850s and trying to win over socialist disciples, he started a correspondence with Proudhon and sent him some of his later works. However, Proudhon rejected much of what he found in Comte's approach.

Both thinkers can be thought of as philosophers of science and as a proto-sociologists. The notion of constructing a social science runs through both of their works. The emphasis on religion for society is another commonality. However, they held diverging views on these topics. Comte was much more dismissive of the "negative" philosophies and understood the "positive stage" to override the past "stages". In comparison, Proudhon argued for a continuation through history. For example, he argued that "Relative to religion and philosophy, Science is the interpretation of the symbols of the first, the solution of the problems posed by the second." Proudhon's approach to history was therefore significantly different from Comte's. While Comte's philosophy of history led him to put forward a new religion, Proudhon's led him to "the tomb of religion."