(Note that the logic systems described in this question *only refer* to logic systems restricted to the language $\in$)

**1. Can $\mathcal{L}_{\kappa,\kappa}$ express $n$-th order finitary logic?** It is clear that it can express first-order logic ($\Pi_{<\omega}^0$), but it seems unlikely that $\mathcal{L}_{\kappa,\kappa}$ can express even $\Pi_{<\omega}^1$.

By the answer to this question: Gödel's Constructible Universe in Infinitary Logics (A Possible Approach to HOD Problem), it is known that $L^{\Pi_{<\omega}^1}=HOD$. Assuming $\mathcal{L}_{\kappa,\kappa}$ expresses $\Pi_{<\omega}^1$, it is then not difficult to show that $HOD\subseteq L^{\mathcal{L}_{\kappa,\kappa}}\subseteq L(V_\kappa)$. Thus, unless $L(V_\kappa)=V$ (which is true iff $L=V$), $V\neq HOD$.

Therefore, the existence of such a $\kappa$ is inconsistent with $V\neq L\land V=HOD$. This is very doubtful.

**2. Can $\mathcal{L}_{\kappa,\kappa}$ express $\Pi_n^1$ for $n>0$?** For $n=0$, it is once again easy to see that, yes, this is true ($\Pi_0^1=\Pi_0^0$) for every $\kappa$. Other than this, I have no information for this question.