John,

all this looks good to me, except:

>  tr(rho^2) is basis-dependent.  (I have
>   specific examples.)
>   

I do not think so.
> * Members of an ensemble apparently do not need to be orthogonal.  I find this
>   bizarre.
>   

Why?
> * Determination of pure vs. mixed, and separable vs. entangled, is utterly
>   trivial for finite-dimensional state spaces.  I understand there are
>   complications in the infinite-dimensional case.
>   
it may be simple at the level of definitions.  But there are nontrivial 
geometric problems coming up in finite dimensions already.
> * It is not at all clear to me how the density-operator representation is
>   in any sense "more general" than the state-vector representation.  One may
>   pass back *and* forth between them.
>
>   
As long as the Hilbert space is kept unchanged, mixed states are more 
general.  This has a physical meaning.

Best regards

Janek