Problems with introducing ordered pairs axiomatically (See also Introducing ordered pairs in an axiomatic way).
Many feel that the usual way to introduce ordered pairs in set theory following K.Kuratowski as 
$(a,b) = \{\{a\}, \{a,b\}\}$ is rather unnatural (B.Russell called Kuratowski's definition a trick).
But the main drawback of this definition is the emergence of so called "junk" theorems - see
https://mathoverflow.net/questions/90820/set-theories-without-junk-theorems
So it seems reasonable to try to introduce ordered pairs in an axiomatic way,
for example, to extend  $ZFC$ by adding to $ZFC$ a new binary functional symbol $g$ 
and an obvious axiom:
Axiom1 := $\forall a,b,c,d( g(a,b) = g(c,d) \rightarrow a=c \wedge b=d)$.
But for further development of set theory we need more axioms, for example
Axiom2 :=  $\forall a,b(g(a,b) \in P(P(a \cup b)))$.
So my question is:
Are necessary some other axioms for the symbol $g$? 
 A: There is no problem introducing a new symbol $g$ for a pairing function to ZFC. Let's call ZFC$g$ the theory with the new symbol $g$ together with the new axiom $$\forall x_1 \forall x_2 \forall y_1 \forall y_2 (g(x_1,x_2) = g(y_1,y_2) \to x_1 = y_ 1 \land x_2 = y_2)$$ and also with the comprehension and replacement axioms expanded to include formulas mentioning this new function symbol $g$.
Since $\newcommand{\ZFC}{\mathsf{ZFC}}\ZFC$ has a definable pairing function (e.g. the Kuratowski pairing function) this new theory is $\ZFC_g$ relatively consistent with $\ZFC$. In fact, since every model of $\ZFC$ can be expanded to a model of $\ZFC_g$, it is conservative over $\ZFC$: every statement that doesn't involve $g$ that is provable in $\ZFC_g$ was already provable in $\ZFC$. In other words, $\ZFC_g$ is a 100% harmless extension of $\ZFC$.
This trick does avoid most junk theorems, e.g. $\varnothing \in g(\varnothing,\varnothing)$ is neither provable nor disprovable in $\ZFC_g$. However, it does not avoid junk facts, e.g. $\varnothing \in g(\varnothing,\varnothing)$ has to be true or false in any model of $\ZFC_g$. As with any conventions for tuples, functions, sequences, and so on, $\ZFC$ is completely agnostic as to how these are defined, it is only the fact that at least one encoding exists that really matters.
A: What you are looking for is known as structural set theory. There are many possible formulation, but usually they do not stop with just having axiomatic ordered pairs. For one example, see the SEAR entry in nLab, as well as Todd Trimble's entries on ETCS (part 1, part 2, part 3). Part I has an explicit axiomatization of ordered pairs, look up "Axiom of products".
Briefly, ordered pairs are axiomatized as follows:

For all sets $a$ and $b$ there is a set $c$ and two functions $p_1 : c \to a$ and $p_2 : c \to b$, such that for all $x \in a$ and $y \in b$ there exists a unique $z \in c$, such that
  $p_1(z) = x$ and $p_2(z) = y$.

I am skipping over a couple of details that Todd properly attends to. Your function symbol $g$ corresponds to the unique existence of $z$ above.
A: I can't say that I approve of this programme, because coding and representation are frequently necessary in set theory, and yield such arbitrary theorems every time. But to answer your question...
You will need a way to distinguish the $g(a,b)$ from other sets. So, in particular, an axiom like $\emptyset\not\in g(a,b)$ seems necessary. Moreover, knowing that g(a,b) is a finite set will be useful.
