Aren't the states of two identical particles always entangled even if they are not interacting? The states of two identical particles are either symmetric or antisymmetric i.e., cannot be written as product states.
Entanglement is only a meaningful concept when there is a well-defined notion of subsystems, which generally means spatially separated subsystems. Indeed, the notion of "product state" (or its converse, "entangled state") is only meaningful relative to a given tensor product decomposition of the Hilbert space, which implicitly defines a splitting into subsystems. Hence, it is not obvious that indistinguishable particles with an (anti-)symmetrised wavefunction can be truly regarded as entangled subsystems.
Nevertheless, Killoran, Cramer and Plenio (arXiv link) showed that this is indeed a genuine form of entanglement. That is, the entanglement formally associated with the symmetrised wave function of a system of indistinguishable bosons can be extracted into the equivalent amount of entanglement (asymptotically) between distinguishable modes. I believe a similar result was found also for fermions by Cavalcanti et al. (arXiv link).
Yes and no. Yes if you look at the formal definition, then yes, but this is not really what is meant. Only the full state in symmetrized, so when you have one particle going to Alice and one particle going to Bob and they perform e.g. a spin measurement, then already by the 'location' of the particle detection the symmetrization is 'voided', and now the question if the spin degrees are entagnled makes sense.