To what extent did progressivism and imperialism recognize and support the same notions and values about America and her place in the world?
They did this to a great extent. Both progressivism and imperialism held that mainstream American culture was in some way special and superior to other ways of life. The progressives manifested this by trying to "improve" people like immigrants and make them more like mainstream, middle class Americans. The imperialists manifested this by doing things like taking the Philippines with the intention of "civilizing" the Filipinos.
There were ways in which progressivism and imperialism were not necessarily similar. Progressivism did not necessarily have a militaristic aspect like that of Alfred Thayer Mahan. However, because they both saw mainstream America as superior to other countries and other ways of life, the two were closely connected.