I get confused when in papers it is said that:
"Something is holomorphic (Complex, symplectic, etc ...) in some Complex structure"
What is the meaning of this in general?
For example in a paper contains stuff about Hyper-Kahler manifold $\mathcal{M}$, it is mentioned that:
$w_3$ (one of the Kahler forms of $\mathcal{M}$) is $(1,1)$ form in complex structure $J_3$ (one of the three complex structures relative to which $\mathcal{M}$ is hyper-Kahler)
or
$\omega$ is a $(2,0)$ form in complex structure $J^{\zeta}$ ($\zeta$ is a coordinate on $\mathbb{CP}^1$ that parameterizes complex structures $J^{\zeta}$ relative to which $\mathcal{M}$ is Kahler)
A complex manifold is a topological manifold equipped with an equivalence class of atlases consisting of charts which have holomorphic transition functions.
An almost complex manifold is a pair $(M, J)$ where $M$ is a smooth manifold and $J$ is a bundle endomorphism $J : TM \to TM$ with $J\circ J = -\operatorname{id}_{TM}$; we call $J$ an almost complex structure.
Every complex manifold admits a natural almost complex structure, but not every almost complex structure comes from a complex manifold; those which do are called integrable. Integrability can be characterised in several ways, the most common is by the vanishing of the Nijenhuis tensor - this is the content of the Newlander-Nirenberg Theorem.
Therefore, one can consider a complex manifold to be a smooth manifold $M$ equipped with an integrable almost complex structure $J$. In this way, $J$ determines the complex structure (and is sometimes just called the complex structure).
So to say an object has some property with respect to (or 'in') $J$ means that the object, when considered as an object on the complex manifold determined by $J$, has the stated property.
For example, in your second statement, $\omega$ is a two form on $M$ (the base manifold), so it is a section of $\bigwedge^2(T^*M\otimes_{\mathbb{R}}\mathbb{C})$. In order to determine it's decomposition into $(2, 0)$, $(1, 1)$, and $(0, 2)$ parts, you need to know how the bundle splits - this is determined by $J$ (in fact, this doesn't require $J$ to be integrable). What is being said is that using the splitting given by $J_3$, $\omega$ is $(2, 0)$-form, but for some other $J$, $\omega$ may not be a $(2, 0)$-form.