1. Yes, the nuts and bolts of the question must be based on relevant and defensible definitions of 'part', 'independent' and such.
2. I don't think you can incorporate one pair of an entangled pair in an atom and retain the entanglement--the interaction will disentangle them.
3. Your...
Some say that there are parts but their properties depend on the whole. But if we cannot assign properties to the 'parts' of an entangled system, what can it be to be a 'part' or any kind of entity in the first place? Can we say that there are parts but that they are not independent? What can...