  We address the thermodynamics, density profiles and superfluid density of
trapped fermions undergoing BCS-BEC crossover, with and without population
imbalance. Our approach represents a fully consistent treatment of "pseudogap
effects", which reflect the distinction between the pair formation temperature
$T^*$ and the pair condensation temperature $T_c$ and must be accommodated by
modifying the fermionic excitation spectrum. This distinction has been omitted
from all other many body approaches in the literature. We show how enforcing
this corollary implies that pairing fluctuation contributions enter into both
the gap and the number equations; this is necessary in order to be consistent
with a generalized Ward identity. Furthermore, we demonstrate that we obtain
physical results for the superfluid density $n_s(T)$ at all $T$. In contrast,
previous work in the literature has led to ill-behaved $n_s(T)$. The superfluid
density is a critical measure of the physicality of a given crossover theory.
In order to properly address thermodynamic properties of a trapped Fermi gas,
it is necessary to demonstrate that the particle density profiles are
consistent with experiment. The lack of demonstrable consistency between
theoretical and experimental density profiles, along with problematic behavior
found for $n_s(T)$, casts doubt on previous claims in the literature concerning
quantitative agreement between thermodynamical calculations and experiment.
