It kind of depends on what the question means by 'simplest', but in terms of number of symbols, you cannot do better: the truth of the statement clearly depends on $p$: if $p$ is true, then you get a different truth-value for the statement than when $p$ is false. Same goes for $q$. So, the statement needs both a $p$ and a $q$ ... and therefore also at elast one operator. So, your statement, that has a single $p$, a single $q$, and one operator is 'simplest' in terms of number of symbols.
An equally 'simple' statement would be $q \leftrightarrow p$ of course.
Finally, is $\neg p \leftrightarrow \neg q$ the contrapositive of $p \leftrightarrow q$? Well, it's a little unusual to talk about contrapositives of biconditionals, as typically this is used for one-way conditionals: the contrapositive of $p \rightarrow q$ is $\neg q \rightarrow \neg p$. But, as such, if there is such a thing as a contrapositive of a bi-conditional, I would expect it to be $\neg q \leftrightarrow \neg p$ ... although that is of course equivalent to $\neg p \leftrightarrow \neg q$.