Yes, especially since it means "man" in its general and slightly archaic sense: humanity. It can also sometimes refer to any person, such as when it's used in the law to refer to anyone who contracts a skin disease (Leviticus 13:2, 9). Unfortunately, our Bibles usually only transliterate the sound of the Hebrew word into Adam in places where it seems to be a name. Elsewhere it's translated as a variety of other words ("people", "person", "mankind", "man", "everyone", "one", "mortals", "mortal"). Even worse, in the one place where Adam is explicitly given as a name -- by God to all the first humans, male and female, in Genesis 5:2 -- modern translations typically use a different name instead. This makes it very easy for readers to not notice that Adam means humanity.
So yes, since Adam means humanity, the Eden accounts seems a bit like Ezekiel 16 where we have a woman named Jerusalem. The name suggests the character tells the story of more than an individual. (Where it gets tricky, of course, is that we don't have Jerusalem's genealogy! But I'm wary of making genealogies the linchpin for how to understand Adam.)
While I'm open to a local flood, I lean towards a parabolic reading due to one feature of the text. A key reason God sends the flood is because "The LORD saw that the wickedness of humankind was great in the earth, and that every inclination of the thoughts of their hearts was only evil continually" (Genesis 6:5). Later, we get the reason there will never be another similar destruction of the earth: "the LORD said in his heart, ‘I will never again curse the ground because of humankind, for the inclination of the human heart is evil from youth; nor will I ever again destroy every living creature as I have done" (Genesis 8:21).
It appears God had the same reason for sending the flood and never sending another one! The account reads as if God thinks the flood is going to fix the problem, but when it doesn't, God gives up. Now, I can accept that as a parable for why God is merciful with us in spite of our evil -- how even if God started over with the best of us, they'd still cause things to go south -- but it doesn't work to me as an accurate portrayal of how God found out how messed up we are. I think God knew that all along. I don't think God ever thought that wiping out all humans save eight would fix that problem.
Yes, though I'd put it more as God giving humanity an undeserved promotion and special role as in Psalm 8 ("what are human beings that you are mindful of them, mortals that you care for them? Yet you have made them a little lower than God, and crowned them with glory and honour") rather than tying it to an immortal soul. That concept seems quite foreign to Genesis, especially due to the purpose of the tree of life and the expulsion from the garden so humans don't become immortal.
Anyway, just some of my thoughts. I appreciate the chance to dialogue!