Let $G$ be a finite group and $${\rm Cl}(x) = \{g\,x\,g^{-1}\mid g\in G\}$$ the conjugacy class of $x\in G$. As explained here, in general, the conjugacy class ${\rm Cl}(x)$ does not necessarily contain the inverse element $x^{-1}$.
My question: does it always contain $x^{-1}$, if $G$ is the symmetric group $S_n$?
On
Yes it does, one reason for that is that the conjugacy classes in the symmetric group is the set of permutation with the same cycle structure, and that a permutation and its inverse have the same cycle structure (see here for instance)
Yes. To see it, all you need to do is note that there's no permutation whose inverse has a different cycle structure.
For it is an important fact that any permutation can be written as the product of disjoint cycles. The inverse is then the product of their inverses. But the inverse of an $n$-cycle is that cycle to the $n-1$-st power. And $(n,n-1)=1$. It's another well-known result that an $n $-cycle raised to a power relatively prime to $n $ is again an $n $-cycle.