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
$$H = \{X − i JX|X \in D\}$$
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 all 1990).

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.