Let $T$ be a complete countable first-order theory. I will write $I(T,\kappa)$ for the number of models of $T$ of cardinality $\kappa$ up to isomorphism. The function $I(T,-)$ is called the spectrum of $T$. The possible spectra are completely understood (for uncountable $\kappa$), see The uncountable spectra of countable theories by Hart, Hrushovski, and Laskowski). For background, I'll sketch what's going on in the cases when some $I(T,\kappa)$ is finite, and then I'll show that the set $X$ is computable.
- As noted in the question, $I(T,\aleph_0)$ can be any natural number except $2$. A theory $T$ with $1 < I(T,\aleph_0) < \aleph_0$ is called an Ehrenfeucht theory. It's a theorem of Lascar that no Ehrenfeucht theory is superstable, and it's a theorem of Shelah that if $T$ is not superstable, then $I(T,\kappa) = 2^\kappa$ for all uncountable $\kappa$.
- Morley showed that if $I(T,\kappa) = 1$ for some uncountable $\kappa$, then $I(T,\lambda) = 1$ for all uncountable $\lambda$. Such theories are called uncountably categorical, and if $T$ is uncountably categorical, then $I(T,\aleph_0) = 1$ or $\aleph_0$.
- By a theorem of Shelah (generalizing Morley, who handled the case of $I(T,\kappa) = 1$), if $\kappa$ is uncountable and $I(T,\kappa)$ is finite, then $T$ is $\omega$-stable. By another theorem of Shelah, if $T$ is $\omega$-stable but not uncountably categorical, then $I(T,\aleph_\alpha)\geq |\alpha+1|$.
- By the last point, if $1 < I(T,\kappa) < \aleph_0$, with $\kappa$ uncountable, then $T$ is $\omega$-stable and $\kappa = \aleph_k$ for some finite $k$. Lachlan built on Shelah's work by showing that in this case, there is some finite $d>1$ such that models of $T$ are determined up to isomorphism by the dimensions of $d$ strongly minimal sets. (This directly generalizes the classification of models in the uncountably categorical case, where the isomorphism type of a model is determined by the dimension of $d = 1$ strongly minimal set.)
More precisely, Lachlan showed that the countable prime model $M\models T$ contains $d$ strongly minimal sets, say with dimensions $(\mu_1,\dots,\mu_d)$ in the prime model, and for any sequence of cardinals $(\mu_1',\dots,\mu_d')$ with $\mu_i\leq \mu_i'$, there is an elementary extension of $M$ of cardinality $\max(\mu_1',\dots,\mu_d')$ in which the $d$ strongly minimal sets have dimensions $(\mu_1',\dots,\mu_d')$, and if two models have the same $d$-tuple of dimensions, then they are isomorphic.
It follows that all the $\mu_i$ are $\aleph_0$ (otherwise, if say $\mu_1$ was finite, we could get infinitely many models of size $\aleph_k$ by allowing $\mu_1'$ to range through the finite cardinals greater than $\mu_1$ and fixing the other $\mu_i'$ at $\aleph_k$), and hence $I(T,\aleph_0) = 1$ (since the prime model is the unique countable model of $T$).
Now the models of $T$ of cardinality less than $\aleph_k$ can be classified by a $d$-tuple of infinite cardinals $< \aleph_k$, which we can code by an element of $k^d$. But there's an additional hitch: the automorphism group of the prime model might act non-trivially on the $d$ strongly minimal sets. So associated to $T$, we have a subgroup $G$ of $S_d$, and the number of models of $T$ of cardinality less than $\aleph_k$ is equal to the number of orbits of the natural action of $G$ on $k^d$ (by permuting the indices of the $d$-tuple from $k$). Further, given any subgroup $G$ of $S_d$, we can cook up a theory whose models are classified in this way: basically, we find a finite structure $A$ of size $d$ whose automorphism group is $G$, and then consider the theory of an equivalence relation with $d$ classes, each of which is infinite, and equip the quotient with the structure of $A$.
All of this is laid out in two papers by Lachlan: Theories with a finite number of models in an uncountable power are categorical, which one you linked to in your question, and Spectra of $\omega$-stable theories. See especially Section 3 of the latter paper for the description in terms of finite permutation groups and the construction of a theory from a group. These papers also contains references to all the facts I mentioned above. See also Theorem IV.10 in Pillay's The models of a non-multidimensional $\omega$-stable theory.
To count the orbits of the natural action of $G\leq S_d$ on $k^d$, we can note that an element of $k^d$ is fixed by $\sigma\in S_d$ if and only if it is constant on each of the orbits of the action of $\sigma$ on $d$. Write $o(\sigma)$ for the number of such orbits. By Burnside's Lemma, the number of orbits of the action of $G$ on $k^d$ is $$\frac{1}{|G|}\sum_{\sigma\in G} k^{o(\sigma)},$$ which is a polynomial $p_G(k)$ in $k$ of degree $d$ (since the maximum $o(\sigma)$ is $o(\mathrm{id}) = d$) with positive coefficients and no constant term. Note also that $p_G(0) = 0$ and $p_G(1) = 1$. Writing $q_G(k) = p_G(k+1)-p_G(k)$, if the group associated to $T$ is $G$, then $I(T,\aleph_k) = q(k)$, which is also a polynomial in $k$ with positive coefficients.
Note that for fixed $k$ and $d$, the group $G$ that gives the fewest models is the entire symmetric group $S_d$. We have $p_{S_d}(k) = {k+d-1 \choose d}$ (the number of ways to make $d$ choices from $k$ elements with repetitions allowed), so $$q_{S_d}(k) = {k+d \choose d} - {k+d-1 \choose d} = {k+d-1\choose d-1} = {k+d-1 \choose k}.$$
This function is increasing in $d$, and it follows that $X$ is computable. To determine whether $(m,k)\in X$ with $k\geq 1$, pick $d$ large enough such that $m < q_{S_d}(k) = {k+d-1 \choose k}$, enumerate all subgroups $G$ of $S_{d'}$ for all $d'<d$, and check whether $q_G(k) = m$ for any of these groups. [And when $k = 0$, $(m,k)\in X$ if and only if $m\neq 2$].
For example, since $q_{S_d}(1)={1+d-1\choose 1}=d$, every natural number $m$ can occur as $I(T,\aleph_1)$ for some $T$ (explicitly, the theory of an equivalence relation with $m$ classes, all of which are infinite - the models of size $\aleph_1$ are determined up to isomorphism by how many classes are uncountable)
Since $q_{S_4}(2) = {2+4-1\choose 2} = {5 \choose 2} = 10$, and $q_{S_4}(3) = {3+4-1\choose 3} = {6 \choose 3} = 20$, to determine which numbers less than $10$ occur as $I(T,\aleph_2)$ for some $T$, or to determine which numbers less than $20$ occur as $I(T,\aleph_3)$ for some $T$, we only have to consider subgroups of $S_3$.
Looking at theories whose associated groups are $S_1$, $S_2$, the trivial subgroup of $S_2$, $S_3$, $A_3$, an order $2$ subgroup of $S_3$, and the trivial subgroup of $S_3$, in that order, we obtain:
- For $\aleph_2$: $1$, $3$, $5$, $6$, $7$, $12$, or $19$ models. (Note that this list is only complete for the numbers less than $10$.)
- For $\aleph_3$: $1$, $4$, $7$, $10$, $13$, $22$, or $37$ models. (Note that this list is only complete for the numbers less than $20$.)
So (assuming I have done my computations correctly), no complete countable first-order theory $T$ has $I(T,\aleph_2) = 2$, $4$, $8$, or $9$, and the only numbers less than $20$ that occur as $I(T,\aleph_3)$ for some countable complete first-order theory $T$ are $1$, $4$, $7$, $10$, and $13$.
The question James raised in the comments, about whether the numbers that occur as $p_G(k)$ or $q_G(k)$ can be characterized in some nicer way that doesn't involve enumerating all finite permutation groups, is also interesting.
