About the constants in the Language of Set theory (with ZF) Accordingly, the language of Set Theory is made up only with the relational symbol "$\in$". But then how to deal with constants? Let's consider for example the case when defining "$\emptyset$". From the axiom of extensionality and the axiom of existence we get to the expression $\exists! X\forall Y(Y\notin X)$. In the first, place what is the meaning of saying  that we define this "$X$" to be $\emptyset$? Also, what is $\emptyset$ if not a constant? What happen when adding more axioms to build a new theory, like say group theory in which we need to use constants?  
 A: I think you are confusing between constants in the language (of sets) and constants in the meta-language. The language of $ZF$ does not have constants. That means that you are not obliged to interpret any constant as a particular set. There are simply no constants to interpret. You only need to interpret the symbol $\in$. Now, it turns out that once that is done it is a theorem that $\exists !X\forall Y \neg (X\in Y)$. That means that in the interpretation there is a unique set $X$ with some property. Now we turn to the meta theory (where we have plenty of constants) and assign this unique set $X$ to the constant $\emptyset$ of the meta language. We then (confusingly perhaps) freely use $\emptyset$ when we argue formally in $ZF$. This is fine as long as you remember that when you see $\emptyset$ used in a $ZF$ formula, you need to replace it (appropriately) by the sentence of $ZF$ that defines what $\emptyset $ is. Once you do that you'll get an honest $ZF$ formula (i.e., with no constants). Of course, this is very tedious and so is avoided. We use definition to avoid writing the same thing again and again and again and again. 
A: Effectively we use a lot more symbols in the language of set theory then what we have in the language.
For example, $\varnothing,\omega,\omega_1,\aleph_\omega,\subseteq,\mathcal P,\bigcup$ and so on. The reason we can do that is that all of these things are definable elements, relations, functions, whatever.
This means that if we add those symbols to the language and add axioms stating that these symbol satisfy a particular formula, then we really didn't change anything substantial in our systems, at least in terms of provability.
So these symbols are actually shorthand for formulas. Much like often we agree that the logic only has $\lnot,\land,\exists$ and the rest of the connectives and $\forall$ are just shorthand for more complicated formulas.
But to your second question, when we want to build the axioms of group theory in $\sf ZF$, we don't add axioms to $\sf ZF$. No, instead we formalize the notion of a language, and logic, and all that and we define a set of symbols with which we can write the axioms of group theory. In that set we can decide to have a symbol for the neutral element (although that too is a definable constant from the axioms of group theory).
