A compact complex manifold $X$ satisfies the Hodge decomposition $$H^k_{\mathrm{DR}}(X, \, \mathbb C) = \bigoplus_{p+q=k}H^{p, q}(X)$$ (possibly without Hodge symmetry) if its Frölicher spectral sequence
$$E_1^{p,q} = H^{p,q}(X) \Rightarrow H^{p+q}(X)$$
degenerates at $E_1$. This happens for instance if $X$ satisfies the $\partial \bar{\partial}$-lemma.

All Moishezon manifolds satisfy it, because the $\partial \bar{\partial}$-lemma is true for projective manifolds, so it is true for any bimeromorphic modification of them. Furthermore, a Moishezon manifold admits a Kähler metric if and only if it is projective.

Thus, in order to obtain the example you are looking for, it suffices to take a non-projective Moishezon manifold (which only exists in dimension $\geq 3$).

For more details and references, you can look at

Dan Popovici, **Deformation openness and closedness of various classes of compact complex manifolds; examples**. *Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)* **13** (2014), no. 2, 255--305.

no, because the classification implies that any compact complex surface with even first Betti number is necessarily a Kaehler manifold. $\endgroup$