I have three three questions, the first two of which probably have the same answer and the third of which is more vague.

For a set $A$ let $L_\alpha(A)$ be the constructible universe up to $\alpha$, built from $A$ as a set (and not a predicate). Further let $X = (B, f)$ where $B$ is a transitive set and $f$ is a bijection from $\omega$ to $B$.

Also assume that the background universe has whatever large cardinals you would like (or that would be helpful). In particular though there is at least one inaccessible cardinal in $L$.

(1) Suppose $L_\kappa\models ZFC$. Is it the case that $\omega_1^L = \omega_1^{L_\kappa}$?

(2) Suppose $L_\kappa(X)\models ZFC$. Is it the case that $\omega_1^{L(X)} = \omega_1^{L_\kappa(X)}$?

(3) If the answer to (1), (2) is yes, is there any simpler way for $L_\alpha$ to know that $\omega_1^{L_\alpha} = \omega_1^L$ (other than $L_\alpha\models ZFC$)?

Finally I will just make one observation to highlight why this question isn't trivial. If you replace $ZFC$ with $KP$ then there are many countable admissible sets $L_\alpha\models KP$ with countable (in $L$) ordinals $\beta\in L_\alpha$ such that $L_\alpha \models \omega_1 = \beta$.

Thanks