Fix the language of set theory $\mathcal{L}=\{\in\}$. Let $\langle M,\in\rangle$ be a set or proper class model of ZFC (e.g. $M$ could be $L$, $HOD$, $V_{\kappa}$ for some inaccessible cardinal $\kappa$, etc.) consider the complete theory of this model in the language $\mathcal{L}$ that is $T_{M}:=Th(\langle M,\in\rangle)=\{\sigma\in\mathcal{L}~|~\langle M,\in\rangle\models \sigma\}$. My questions are about the possible number of countable models of this complete theory up to isomorphism $I(T_{M},\aleph_0)$ when $M$ varies over different models of $ZFC$.
Question 1: What are $I(T_{L},\aleph_0)$ and $I(T_{HOD},\aleph_0)$, $I(T_{V_{\kappa}},\aleph_0)$ (for $\kappa$ inaccessible)?
Question 2: Is there a set or proper class model $M$ of ZFC such that $I(T_{M},\aleph_0)=\lambda$ for each cardinal $\lambda\in\{1,3,4,\cdots,\aleph_{0},\aleph_1,2^{\aleph_{0}}\}$?
Levon Haykazyan has the right idea with his comment. Any consistent completion of ZFC (call it ZFC$^*$) has $2^{\aleph_0}$ countable models up to isomorphism.
Note that relative to ZFC, each finite ordinal is definable without parameters. $0$ is definable by $\phi_0: \forall y\, \lnot y\in x$, and $n+1$ is definable by $\phi_{n+1}:\forall y\,(y\in x)\leftrightarrow \lor_{i\leq n} \phi_i(y)$.
Now for each $S\subseteq \omega$ (our $\omega$!), consider the partial type $$p_S(x) = \{\exists y\, \phi_n(y)\land y\in x\mid n\in S\}\cup\{\lnot \exists y\, \phi_n(y)\land y\in x\mid n\notin S\}.$$ This is a family of $2^{\aleph_0}$ pairwise contradictory partial types over $\emptyset$, all of which are consistent with ZFC$^*$. Each of them can be realized in a countable model of ZFC$^*$, but any countable structure only realizes countably many types over $\emptyset$, and any two isomorphic structures realize the same types. So ZFC$^*$ must have $2^{\aleph_0}$ countable models up to isomorphism.