Physical meaning of NCP (not completely positive) maps 1) Even with not completely positive maps, if we can find a domain in which the action of the map is positive, why should we restrict the reduced dynamics of an open system to being exclusively completely positive? Also, what could such maps indicate physically about the system?
2) What are the arguments put forward to justify completely positive maps for describing reality? Is it always possible to decouple the system from the environment and vanish the correlation terms to ensure the evolution of the joint state to be CP (completely positive)?
 A: If a map can be applied to unentangled inputs, e.g. the first qubit in $|0\rangle|0\rangle$ and $|1\rangle|1\rangle$, linearity implies that it can also be applied to $|0\rangle|0\rangle+|1\rangle|1\rangle$.  Thus, you can only restrict the input domain of your map (e.g. such as to ensure positive output of a non-CP map) if you are willing to give up linearity. 
A: If a map $\mathcal E$ is NCP, then it will produce unphysical states for some input states that are correlated with other "external" systems.
Therefore, either you say that it is not possible (as in, it makes no physical sense) for that map to act on those states, or QM is to be changed to make sense of non-positive output states.
The former case is also weird though, because the restriction on the input state is really a restriction over the possible extensions of such states.
In other words, you are not saying that $\mathcal E$ cannot act on $\rho$ for whatever reason, but that $\mathcal E$ can act on $\rho$ as long as $\rho$ is not correlated with something else.
Physically speaking, the existence of an NCP map would mean that you can build a box such that when you put in states correlated with some external systems, you get output states which on a first glance may appear to be proper states (as the output reduced state is still a physical state), but that when you look more carefully (that is, you look at the global output state, taking into consideration also the other system the input was correlated with) you realize has produced something which makes no sense.
Note that while this means that NCP maps are not physically meaningful maps, they are still of the utmost importance, among other things, for the theory of entanglement. Indeed, it is a standard result that a state is separable if and only if its positivity is preserved by all positive maps. In this sense, the separability problem is equivalent to the classification of all positive maps.
