Your problem is that your intuitions about the empty set don't fit the mathematical definitions; where functions etc. end up existing vacuously. In this answer, given your background I'll discuss the category Set. This category has as its objects sets and as its morphisms functions between sets.
In this setting, your question asks what the morphisms with domain $\emptyset$ are. The point is that the definition of a function $f:A \to B$ is "for each $a \in A$ there is a unique $b \in B$ such that $f(a) = b$" (In set theoretic language $f:A \to B$ is actually a subset of $A \times B$ such that for each $a \in A$, there is a unique $b \in B$ such that $(a,b) \in f$).
In particular, this definition means that for any set $B$ there is a unique function from $\emptyset$ to $B$; namely $\emptyset$ itself! (though in the category we distinguish between the object $\emptyset$ and each of the morphisms $\emptyset$ since the morphisms come with a domain and codomain) This may seem strange, but the definition requires it since there are no $a \in \emptyset$ so the needed condition vacuously holds for $f = \emptyset$. In particular, in Set for every object $B$ there is a unique morphism $f:\emptyset \to B$ (In technical language, this says that $\emptyset$ is an initial object in Set). So, yes, in Set, $\emptyset$ has an identity morphism $\emptyset \to \emptyset$ and also participates in morphisms to all other sets. This means your intuition about the category Set is incorrect.
In general, a category is really just a collection of objects with arrows (morphisms) satisfying the right rules. Anything you write down with those properties is fine! We could consider the category whose objects are sets with just the identity arrows, or with all possible arrows; both would be fine. The advantage of this is that it lets lots of things fit into this framework.
The real question here is what properties the language of category theory captures.
A statement in the (finitary) language of category theory is one formed from propositions of the form $f = g$ where $f, g : X \to Y$ using the propositional operations of $\lor, \land, \top, \bot, \neg, \implies$, together with quantifiers either over all objects (eg $\forall X$, $\exists X$ where $X$ is an object variable) or over all morphisms between two given objects (eg $\forall f : X \to Y$ or $\exists f : X \to Y$ where $X, Y$ are objects).
Two things can be shown about statements in the language of category theory. Suppose given a statement $\phi(X_1, ..., X_n, f_1, ..., f_m)$ in the language of category theory with object variables $X_1, ..., X_n$ and free function variables $f_i : X_{d_i} \to X_{c_i}$ for $1 \leq i \leq m$.
Now consider a category and two different variable assignments - one assignment $X_1 \mapsto W_1, ..., X_n \mapsto W_n$ where each $W_i$ is an object, and $f_1 \mapsto g_1, ..., f_m \mapsto g_m$ where $g_i : W_{d_i} \to W_{c_i}$ for all $i$, and another assignment $X_1 \mapsto Y_1, ..., X_n \mapsto Y_n$ and $f_1 \mapsto h_1, ..., f_m \mapsto h_m$, where $Y_i$ is an object for all $i$ and $h_j : W_{d_j} \to W_{c_j}$ for all $j$. Suppose there are isomorphisms $k_1 : W_1 \to Y_1, ..., k_n : W_n \to Y_n$ such that for all $i$, $h_i = k_{c_i} \circ f_i \circ k_{d_i}^{-1}$. Then $\phi(W_1, ..., W_n, g_1, ..., g_m) \iff \phi(Y_1, ..., Y_n, h_1, ..., h_m)$. This is known as "isomorphism invariance of truth".
Now consider a functor $F : C \to D$ which is fully faithful and essentially surjective, and a variable assignment in $C$ of the form $X_i \mapsto W_i$, $f_j \mapsto g_j$ as above. We assume that $\phi$ has no free variables other than the $X_i$ and $f_i$. Then $\phi(W_1, ..., W_n, g_1, ..., g_m)$ iff $\phi(F(W_1), ..., F(W_n), F(g_1), ..., F(g_m))$. This is known as "equivalence invariance of truth".
Both of the above can be proved by induction on formulas.
The two above statements can be generalised to the infinitary language of category theory, which allows quantification over external sets, though some care must be taken in settings without the axiom of choice.
So the language of category can only discuss properties which are invariant up to isomorphism and up to equivalence of categories. No finer-grained discrimination is possible.
For example, consider the category of sets. It follows from isomorphism invariance that the language of category cannot distinguish two sets which have the same cardinality. The language of category cannot distinguish a monic $f : S \to T$ from a subset $S \subseteq T$.
For the category of topological spaces with morphisms the continuous maps quotiented by homotopy, the language of category theory cannot distinguish between homotopy equivalent spaces.
So the key when using category theory is to find the right amount of data for morphisms to carry based on the underlying subject one is studying and the particular context.
Best Answer
What are the "objects of a language"? The entities its first-order quantifiers run over? [That's the usual story!]
Category theory is usually set up with a two-sorted first order language with objects and morphisms as distinct sorts. But it would only be mildly perverse to take the official language of category theory to be single-sorted, with quantifiers running over entities comprising the category's objects and the morphisms (then restricting quantifiers when you want to talk about just the objects, or just the morphisms).
In this single-sorted version, the objects in the single domain of the language will include morphisms between [some of] the domain's own objects.
Presumably, though, you are not interested in this triviality! So I guess the question needs to be refined to give us something clearer to grapple with (so we know whether slice or arrow categories are relevant to real your concerns, for example.)