There are too many Wick's Theorems!

Various comments to the post (v3):

1. One may speculate that seemingly unordered operators are in practice always ordered wrt. some order.

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3. As long as the fields $\phi_i=\phi_i^{(+)}+\phi_i^{(-)}$ are linear in creation and annihilation operators, this should not be a problem.

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5. Isserlis' theorem is related to the path-integral formulation of Wick's theorem, cf. e.g. this Phys.SE post.

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The single-most important generalization of the operator formulation of Wick's theorem (as compared to my Phys.SE answer) is to consider contractions that doesn't belong to the algebra center. This is often used in CFT, see e.g. Ref. 1.

References:

1. J. Fuchs, Affine Lie Algebras and Quantum Groups, (1992); eq. (3.1.35).