Working on editing this now. I am tempted to say it is a mess, but at least the part I have edited so far is an intelligent discussion of the issue. I will try to edit out anything that's a clear misstatement or unnecessarily confusing.
Listening causes me to ask:
Does Joshua "exist?" Let's presume the answer is yes.
Is Joshua's existence a "property" of Joshua's atoms?
I think the answer is that it depends on what we mean by property, but if we drill down into what Epicurus was talking about and try to find a bright line between "properties" and "qualities / accidents / events," I think that probably the best wording is that Joshua is a "quality" of his atoms coming together into a body of particular configuration.
I don't read this as saying that individual atoms have qualities --- only after they come together into bodies do "qualities" exist.
Is it correct to say that a "body" has "properties"? It would seem so, because I think in Lucretius there are examples of using the term property to describe attributes that can't be removed from a thing without destroying its essential nature.
I think what I am getting at here is that it would probably help to try to agree on a terminology so as to explain all this:
1 - Something to the effect that only atoms have eternal "unchanging properties;"
2 - But that when atoms come together to form bodies they have "qualities"
3 - That bodies can also have "properties" that are not changeable without destroying the nature of the body.
4- And that bodies also have "qualities" that do change according to context, but with the qualities limited in number of possibilities by the properties of the bodies, which themselves are tied to the properties of the atoms that gave rise to them.
No doubt this is convoluted and needs substantial revision, but I think these issues are what the episode and this section of the text are really about.
These considerations seem to be the determinants of whether we should regard something as "existing" or "possibly existing" versus the bright line that separates them from that which is "impossible to exist."
The overall picture still seems to me to be a discussion of how we use the theory of atoms and void to make sense of the world around us and to separate "what does or can exist." from "that which is impossible to exist." Which may sound simple but would have significant implications on everything else from theory of knowledge to religion to life after death etc.