Take a very general hypersurface $j \colon X \hookrightarrow \mathbb{P}^n$ of sufficiently high degree. Then it is well-known that $X$ is Kobayashi hyperbolic (this was proven by Siu, if I remember correctly).
On the other hand, Lefschetz hyperplane theorem says that if $n \geq 4$ the natural map
$\pi_1(X) \stackrel{j_*}{\longrightarrow} \pi_1(\mathbb{P}^n)$
is an isomorphism. Then $X$ is simply connected, in particular its fundamental group is amenable and so $X$ cannot be Kahler hyperbolic.
This example also satisfies your second request. If fact, again by Lefschetz theorem, see for instance [Dimca, Singularities and topology of hypersurfaces, Theorem 2.6 p. 151], we also have an isomorphism
$\mathbb{C} \cong H^2(\mathbb{P}^n) \stackrel{j^*}{\longrightarrow} H^2(X)$.
By Hodge decomposition, this implies
$H^{2,0}(X)=H^{0,2}(X)=0, \quad H^{1,1}(X)=\mathbb{C}$.