An element $x/p^k$ of the localization of $\prod_n \mathbb Z/p^n$ is zero if $p^{n-l}$ divides $x_n$ for all $n$ and some $l$. Thus it is a zero divisor if the power of $p$ dividing $x_n$ is unbounded. So a regular element is one where the power of $p$ dividing $x_n$ is bounded, say by $m$. Then there exists some $y$ such that $x_ny_n=p^m$ everywhere, so $xy=p^m$, so $x^{-1}=yp^{k-m}$. 

So the localization is classical.

Other questions: I (currently) have no idea.