First a remark to clarify the terms: LDA (local density approximation) is an approximation used within the density functional formalism, which yields the exact solution of our many-body problem as long as we would know the exact functional. In contrast, DMFT (dynamical mean-field theory) is a concept to solve periodic (lattice) Hamiltonians with local interactions. So in fact, the double counting depends on the local interaction (in DMFT) as well as on the employed density functional.
Second: double counting is not specifically arising for DFT+DMFT but it is a general problem if one tries to combine DFT with a many-body correction (i.e. DFT+U ...). The double counting correction you posted looks like the "atomic limit" approximation (see M.T. Czyzyk, G.A. Sawatzky, Phys. Rev. B 49 (1994) 14211–14228).
To your questions:
(1) + (2) Usually "correlated" orbitals are "defined" (either by projection or by a suitable basis set) prior to the calculation, so that the correction of the DF functional by +U/DMFT is limited to a subspace (e.g. 3d or 4f-type electrons) of the original Hilbert space. This subspace is defined by projection operators which are applied to the DF orbital set. Your double-counting correction is derived from the local Hamiltonian (I suspect it is the (mean-field?) Kanamori Hamiltonian, otherwise the average of U and J has no meaning) and therefore it is also written in the basis of "correlated" orbitals. So in the end, $N^l$ as well as $N^l_\sigma$ are evaluated after the DMFT cycles (self-consistent DMFT calculation) with the basis set used in the DMFT cycles (projected orbitals).
Due to the projection operator, you have a well-defined relationship between your DFT and +U/DMFT basis set and thus you can evaluate the total energy functional in the original DFT basis.
Further information (reviews on DFT+DMFT):
- G. Kotliar et. al., Review of Modern Physics Vol. 78, 866-951
- A. Georges, arxiv:cond-mat/0403123
p.s. a short remark at the end: those functionals are not anymore sole density functionals, but orbital-dependent density functionals