If we define a real valued function $\Vert \cdot \Vert$ on the $n^{th}$ order symmetric group $S_n$ satisfying following conditions
$$\begin{align}
& \|x\|=0\iff x=\omega\,\,\,(\text{identity permutation})\\
& \|x\|=\|x^{-1}\|\,\,\,\,\,\forall x\in S_n\\
& \|x*y\|\le\|x\|+\|y\|\,\,\,\,\,\forall x,y\in S_n \\
\end{align}$$
then $\|.\|$ is a norm on $S_n.$
Also $\color{Blue}{d_1(x,y)=\|xy^{-1}\|}$ and $\color{Green}{d_2(x,y)=\|x^{-1}y\|}$ becomes two norms on $S_n.$
(similar to vector norms.)
Hence using a norm function ( function satisfying above conditions) we can convert any finite group into a metric space.
For example; following trivial norm function induced trivial metric on $S_n.$
$$\left.
\begin{array}{l}
\text{if $x\not=\omega $ :}&1\\
\text{if $x=\omega $ :}&0
\end{array}
\right\}
=\|x\|$$
My questions are
1. Does there exist any non-trivial examples norm functions ?
2. Can we find an explicit example ?
A function satisfying the axioms you listed is called a length function.
For $S_n$, the word length would be an example.
The symmetric group $S_n$ is generated by elements $s_1, \ldots, s_{n-1}$, where $s_i$ is the transposition $(i,i+1)$. Define the word length of an element $w$ of $S_n$ to be the smallest $\ell$ for which there is a decomposition $$ w=s_{i_1}s_{i_2}\cdots s_{i_\ell} $$ for some $i_1, \ldots i_\ell\in \{1, \ldots, n-1\}$. Define the word length of the identity permutation to be $0$.
Then the three axioms listed in your question are satisfied by the word length. Thus it is an example of a non-trivial norm on $S_n$ (except for the cases $n=1$ and $n=2$; in the first, all norms are trivial, and in the second, all norms are multiples of the trivial norm).