There are several interrelated questions here, which is appropriate, because Progressivism consisted of many different interrelated strands. The second and third questions are especially important to understanding the Progressive movement, because many while Progressives saw science and expertise as the key to improving society, they also tended to be comfortable, middle-class white people who sought to impose their morality on others, especially working-class men and women.
The best example of this aspect of Progressivism, one which embodies the different strands of Progressivism was Prohibition, what has been called "the noble experiment." Progressive doctors and public servants emphasized the very real dangers of alcohol abuse, which including health hazards as well as the negative effects that it had on families. Women were especially attracted to the movement, including such figures as Carry Nation, described by Michael McGerr as a "respectable Christian woman in her sixties." Her involvement was typical of the religious, evangelical urge to bring about a heaven on earth. This desire also typified a previous generation of reformers, those who argued for temperance and other reforms before the Civil War. Other people had an interest in regulating the behavior of working-class men, particular business owners, though as McGerr observes, for all the "big-business men who wanted to encourage more temperance and self-discipline in the workforce," there were wealthy men who made money financing the saloons and brothels decried by the prohibitionists. It was also the case that Prohibition marked an unsuccessful attempt to use the power of government to regulate the behavior of Americans. It failed because it was unenforceable and ultimately not consistent with the desires of ordinary Americans. In this sense, it can be argued that this particular aspect of the Progressive movement—and temperance was embraced by even Jane Addams, on the left wing of the movement—was an attempt at social control. Other aspects of the movement, including local government reform and education initiatives, had similar motives, combining an earnest concern for the conditions that faced American working-class people with a middle-class assumption that they knew what was best for "the people."
So, while many Progressives also sought to expand democracy, restrain the influence of corporations, and other laudable projects, they also believed that their vision of America was one to which both working-class people and industrialists needed to conform. In the end, it reshaped American society by redefining the relationship between citizens and government as well as the mutual obligations shared by Americans.