I think the answers you are looking for are in this paper by V. Tosatti, see in particular Proposition 1.1, point (4) and Proposition 1.3.
Warning: the holonomy is computed with respect to the Chern connection of the hermitian metric, which is, since $(X,\omega)$ is not necessarily Kähler, not equal in general to the Levi-Civita connection of the underlying riemannian metric!
Have a nice reading!