When Whitman uses the first person, he does indeed seem to mean himself, Walt, "thirty-seven years old in perfect health." However, that is not all that he means by the first person, since Whitman's poetry is also channelling a kind of universal "I" that integrates poet, reader, and indeed all...
of humanity. Whitman is very clear about this: "every atom belonging to me as good belongs to you," he declares in the third line of "Song of Myself." While the poem is most definitely Whitman's "song," it is also just as definitively everyone else's. For Whitman, there is a universality in the particular that can be hard to follow, but understanding this is really key to "getting" what he is about.
Your question about the body is another great example. Sex is another expression of this particular/universal dichotomy that characterizes so much of his poetry. Whitman very much tries to appear "bodily" before the reader, as a witness to the things he describes in his poetry; however, another interpretation is that he also emphasizes the body because he would have the reader as his lover. The most famous example of this is in section five, where Whitman remembers, "I mind how once we lay such a transparent summer morning, / How you settled your head athwart my hips and gently turn'd over upon me / And parted the shirt from my bosom-bone, and plunged your tongue / to my bare-stript heart." Here, the question becomes who the second person refers to: it could be a real lover, it could be the reader, or it could be the spirit of poetry itself. It is the peculiar charm of Whitman (and perhaps his main point) that the precise answer to that question doesn't matter.
Your questions about democracy and American identity cover a lot of ground. I think a basic answer would be that Whitman redefined the idea of "democracy"—that is, he understood democracy not as a political system but as a kind of expression of universal spiritual equality. Whitman famously conflates the term "democracy" with the word "en masse"—for him, "democracy" comes to mean a kind of divine collective consciousness of the people, the sum of all the particularities of our individual lives. His vision of America is consequently comprehensive and radically inclusive. Good and bad, rich and poor, the living and the dead, all contribute to a vision of America that is the sum of all its activities and desires. America is, as Whitman himself said, "the greatest poem."