The key difference between E/E and seeker or traditional churches is the E/E argument that postmodernism requires a different presentation and style of engagement in mission because postmoderns think and respond to things differently than "moderns" do. The church needs to adjust to the changing times is the emphasis. Virtually any work on E/E walks through the differences in the approaches of the two cultural forms. So the argument is that the cultural reality means we must be prepared to contextualize the gospel appropriately for the new era. They do generally claim that neither seeker churches (which are mostly "modern" in approach, especially in size and worship style) nor traditional churches (which are also rooted in different "modern" assumptions than seeker churches) do what is necessary for the new era. There is too much concession to Western culture in the "modern" approaches that exalt passive (receiving) worship, size and efficiency.

Your question about whether postmodern assumptions are embraced is not really the key point (though many in the E/E movment do). The point is that postmodernism is where the audience is and the church needs to be sensitive to that. Thus, it is no critique of the E/E movement simply to try to tie it to postmodernism. Questions include how does that relationship work and is the cultural analysis on target. Again each element of the articulation of cultural differecne and what it means needs to be discussed, one point at a time.

The answer to your final question also depends on who in the E/E movement is in view. Some hold to core doctrines, while the position of some others is less than clear. One of the problems here is the frequent contrastive rhetoric many in the E/E use. So one hears things like "postmoderns like narrative or story, not doctrine and so we offer story and narrative." It can sound more harsh than it is, although in some cases it appears to mean exactly what it implies. Some who speak and write seem to like to express themselves ambiguously on such points or with such exaggeration, so that clarity is lacking on such questions. This is yet another reason to be careful in discussing the movement as a whole. There is a spectrum here, not a tight box. It often takes looking at the whole of what is said, not a few sentences or sound bites, to tell what is believed.