TITLE: Effect of large cardinals on the value of $\omega_1^L$ in $L$
QUESTION [7 upvotes]: 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 $L_\alpha\models ZFC$. Is it the case that $\omega_1^L = \omega_1^{L_\alpha}$?
(2) Suppose $L_\alpha(X)\models ZFC$. Is it the case that $\omega_1^{L(X)} = \omega_1^{L_\alpha(X)}$?
(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

REPLY [4 votes]: Andres Caicedo has answered Questions 1 and 2 as stated, but, just in case you intended $\kappa>\omega_1^L$ in Question 1 (since that's obviously necessary for the proposed conclusion), let me add that this is also sufficient for an affirmative answer to Question 1.  The point is that, if an ordinal $\alpha$ is countable in $L$ then a bijection between $\alpha$ and $\omega$ appears in the construction of $L$ well before stage $\omega_1^L$.  This is essentially contained in Gödel's proof of GCH in $L$.  It uses much less than the full strength of ZFC in $L_\kappa$; you just need enough set theory to talk sensibly about countability of ordinals.