Modal logics that include the necessitation rule are called normal. Not all modal systems are normal. Some of the non-normal logics are S1, S2 and S3, which are consistent with the necessitation rule, and S6, S7, S8 and S9, which are inconsistent with it.
The necessitation rule has the consequence ⊢¬◇⊥. In effect it excludes logically impossible worlds from being accessible to any world. You are correct to say that this has the consequence that logical necessity entails metaphysical necessity.
Your proposed system consisting of K + □(P & ¬P) would be consistent. In fact there is a modal system that Hughes and Cresswell call Ver that is simply K + □P so this would include the theorem □⊥ or □(P & ¬P) as a special case. As you say, without axiom T, □⊥ does not entail ⊥, so this system is not inconsistent. Its frame condition is a single world with no possible worlds accessible to it at all, not even itself. It is a normal modal logic though, since □P holds for all P, including theorems of the underlying logic, so the necessitation rule holds.
In the system Ver, □P holds for all P, even if P itself does not hold. Even □⊥ holds. This makes Ver rather weird. It would not be suitable for interpreting □ as 'necessarily'. Systems like Ver and Triv, which is K + (P → □P) + (□P → P), are not really meant to be useful modal logics. They are usually included in taxonomies of modal systems for the sake of completeness or for pedagogical purposes.
There are non-normal modal logics that allow for impossible worlds. These permit possibilities that lie beyond what is logically necessary. See, e.g. the SEP article on Impossible Worlds.
Hughes and Cresswell, A New Introduction to Modal Logic, 1996.
