For a contact metric manifold $M$ we observe that $J = \varphi_{\vert D}$, i.e. the restriction of $\varphi$ to the contact distribution, defines an almost complex structure on $D=\ker\eta$. Then the associated almost CR-structure of $M$ is given by the holomorphic subbundle $$H = \{X − i JX\mid X \in D\}$$ of the complexification $TM^{\Bbb C}$ of the tangent bundle $TM$. We say that the associated almost CR-structure is integrable if $[H,H] \subset H$. This is equivalent to $[J, J ](X, Y ) = 0$, for any $X, Y \in D$, where $[J, J ]$ denotes the Nijenhuis torsion of $J$. It is known that the associated CR-structure of a three dimensional contact metric manifold is always integrable (S. Tanno 1989).
A contact metric manifold $M$ such that its associated almost CR-structure is integrable will be referred to as a contact strongly pseudo-convex CR-manifold.
It should be noted that every $(\kappa,\mu)$-manifold is a contact strongly pseudo convex integrable CR-manifold (D. E. Blair et al Contact metric manifolds satisfying a nullity condition doi:10.1007/BF02761646).
And answer of your questions:
A) By above comments, every $(\kappa,\mu)$-manifold is both a contact manifold and a CR manifold.
B) In Mitric [1991] and Tanno [1992] it was shown that the tangent sphere bundle with its standard contact metric structure is a CR-manifold if and only if the base manifold is of constant curvature. Thus if we choose $M$ with non-constant sectional curvature then the tangent sphere bundle $T_1M$ with its standard contact metric structure is a contact metric manifold that not a CR-manifold.
For (C) I do not have an example in mind.
