Dear Noson,
regarding the notion of structure, there are various, not always completely overlapping, definitions. To me, structure is essentially what a system and its model have in common---basically, what makes another system a model of some object system. But it's more usual to phrase this in terms of relations, where 'structure' is essentially the set of relations that are born by the elements of some area of discourse (although I have recently come to be skeptical of this way of putting it).
The usual example I like to give is the set of your direct maternal ancestors, ordered by ancestry, and the set of books on your shelf, ordered by thickness. You can use the books as a model of your ancestors, with a book mapping to a more senior ancestor if it is thicker than another. The structure both share is then just the linear order both sets support. This is what allows you to take elements of one set, and draw conclusions about the other: if 'Moby Dick' is thicker than 'The Old Man and the Sea', and 'Moby Dick' maps to Ethel, while 'The Old Man...' maps to Mabel, you know that Ethel must be an ancestor of Mabel.
Granted, it's not terribly elegant, but I think it gets the point across. For another example, take an orrery and the solar system: the orrery models the solar system because of the arrangement of the little metal beads representing the planets, and because of the translation between the gears---all of which are structural facts about the system.
I don't believe (although some do) that this structure has any existence beyond its being instantiated in concrete objects---so this is rather more an 'Aristotelian' than a Platonic picture.
An important fact about structure is that merely specifying a structure doesn't suffice to fix the objects that bear it. This is the famous objection the mathematician Max Newman raised against Russell's philosophy of science, who held that all we know about the world is its structure (causal structure, to be exact). Newman argued that if that's the case, all we know about the world is its cardinality, since any set of the right cardinality can be considered to bear a given structure.
I essentially (and without further extending this) think that this is half the answer, with our knowledge of structure being impressed upon us by the external world (which is, again, a very Aristotelian 'hylomorphic' notion), but with our knowledge of the intrinsic properties of our minds supplying the necessary bearers of that structure (I have just had a paper accepted where I defend this sort of theory).
As for entanglement, yes, my aim was to agree with your view, by defending it against an argument that's sometimes raised to promote a kind of holism---an object-hood going beyond the sum of its parts---via entanglement. The argument, essentially, is that an entangled object is an irreducible whole, as its parts do not suffice to reconstruct it; incidentally, the same argument is also sometimes used to promote a notion of 'bare structure' or 'relations without relata'. Hence, whether an entangled object is just a collection of parts, or a unified whole, is then, according to that argument, not in the eye of the beholder.
I think this argument fails, however, because it intrinsically depends on a choice of tensor product structure, which is not provided by the bare Hilbert space of the system.
Anyway, lest I write another essay in response right here, let me just say I look forward to reading your comments on my essay... After all, it owes quite a bit to your own work!
Cheers
Jochen