Gödel's constructible universe ($L$) is defined using definable power set operator in first order logic ($\mathcal{L}_{\omega ,\omega}$). One can produce such a universe in infinitary logics in the same way using corresponding notions of formulas and definability. Obviously $L$ becomes larger when the logic has more expression power.

For each cardinal $\kappa$ define $L_{\kappa}$ to be Gödel's constructible universe in the infinitary logic $\mathcal{L}_{\kappa,\kappa}$ and $L_{\infty}$ is Gödel's constructible universe in $\mathcal{L}_{\infty,\infty}$.

(1) Is $L_{\kappa}$ a model of $ZFC$ for each cardinal $\kappa$? What about $ZFC+GCH$?

(2) What is $L_{\infty}$?

(3) Is there a (possibly large) cardinal $\kappa$ such that $L_{\kappa}$ is Dodd-Jensen core model, $L[U]$, $HOD$, etc?

(4) What are the consistency strengths of the existence of non-trivial elementary embeddings from $\langle L_{\kappa},\in\rangle$ to itself for different $\kappa$s in the sense of infinitary logic $\mathcal{L}_{\kappa,\kappa}$?

Note that by Prof. Hamkins' answer for $L_{\infty}$ finally it reach Kunen's inconsistency but what about a given cardinal $\kappa$? Are all these consistency strengths for different cardinals bounded by some large cardinal axiom and there is a gape between consistency strength of the existence of a non-trivial elementary embedding from $\langle L_{\infty},\in\rangle$ to itself and consistency strengths of the existence of such elementary embeddings for $L_{\kappa}$s?

(5) If there is a cardinal $\kappa$ such that $L_{\kappa}=HOD$, is it possible to determine consistency strength of the existence of a non-trivial (first-order) elementary embedding from $\langle HOD,\in\rangle$ to itself by analyzing the growth speed of the consistency strength of existence of such embeddings for $\langle L_{\lambda}, \in\rangle$s in $\lambda <\kappa$?

(6) What is $L_{\kappa}$ for the least strongly compact cardinal $\kappa$?