How do I think of groups as a universal algebra I've been trying to understand universal algebras recently and something in the definition came up which confuses me with respect to how the inverse property should be phrased in this language. 
A universal algebra  is a set of operations $\Omega$ and equivalences $E$ which dictate it type, it also has another set $S$ which are the elements in the algebra. Each operator $\omega \in \Omega$ has an integer associated to it which states its arity, so if this is n then the operation is a function $\omega: S^{n}\rightarrow S$ (assuming some action is defined). These are freely generated by a set of n-ary operators which can essentially be 'plugged' into each other. Finally the equivalences are pairs of operators of the same arity which are said to be the same.
Now defining a group in this language we have a binary multiplication operator $-\cdot-$, a nullary operator, the identity $e$, and a unary operator $(-)^{-1}$ which is the inverse. Finally we need to define equivalences, the associativity and identity axioms seem fine to me (equivalences of 3-ary and 1-ary operators) but the group inverse seems to be an issue.
This axiom is $x \cdot (x)^{-1} = e = (x)^{-1}\cdot x$ which appears to have a 2-ary operator on one side and a 0-ary operator on the other. How does that fit into being viewed as an equivalence between two operators of the same arity? Further the variable x appears twice in this, does this mean we have somehow assumed the ability to copy variables in this framework?
 A: I would recommend talking about the equivalences as being between terms that are built out of operators and variables. Further, I very strongly recommend thinking about terms as being terms with a prescribed set of variables. In other words, if I give you a set, $V$, then I can make the set $T_\Omega(V)$ of terms with variables indexed by the elements of $V$. For example, $v_x\cdot(v_y\cdot v_z)$ is a term of $T_\Omega(\{x,y,z\})$ where $v_x$ (and so forth) is the variable indexed by $x\in V$. An axiom (with variables in $V$) is then a pair of terms of $T_\Omega(V)$ that we are going to forcibly identify via quotienting by the congruence generated by the axioms. A term in $T_\Omega(V)$ does not need to involve all the variables in $V$, so both $e$ and $x\cdot(x)^{-1}$ are terms in $T_\Omega(\{x\})$.
Your very last question about multiple occurrences of the same variable gets into interesting territory that's probably best understood from the perspective of category theory, and specifically categorical logic. Suffice to say, the usual framework we use to do universal algebra fits into the framework of Lawvere theories (and its multi-sorted generalizations). In this framework, we have enough structure to define duplication and ignoring of parameters. (In particular, that structure is having finite categorical products.) We can consider other structures, namely (symmetric) monoidal categories. In this framework, we would not be allowed either to duplicate or ignore variables, so neither $x\cdot(x)^{-1}$ nor $e$ would be valid terms of $T_\Omega(\{x\})$ in this framework. In the jargon of categorical logic, we would say that groups are expressible in the doctrine of categories equipped with finite products but not in the doctrine of a (symmetric) monoidal category. Monoids, on the other hand, are expressible in the doctrine of a monoidal category (and thus the doctrine of categories with finite products) because every axiom uses every variable exactly once. 
The doctrine of (symmetric) monoidal categories is intimately related to substructural logics and in particular linear logic. That said, as far as I can tell most mathematicians are completely unaware of substructural logics, so you will not often come across it unless you study certain fields such as proof theory, formal logic, various areas of computer science, and some areas of theoretical physics. (See e.g. the Rosetta stone paper.) Substructural logics are becoming more broadly known though.
