All pairwise shortest paths in a graph: does knowing the path weights help?
Yes, given the lengths of the shortest paths it is easy to actually find these paths. Suppose for example that you want to find the shortest path from $s$ to $t$. Let $d(s,t)$ be the length of the shortest path, and $w(s,t)$ the length of the edge from $s$ to $t$, or $\infty$ if there is none.

If $d(s,t) = w(s,t)$ then the shortest path is simply $s,t$. Otherwise, there must be a vertex $x$ such that $d(s,t) = w(s,x) + d(x,t)$ (why?). The shortest path then starts $s,x$ and continues with the shortest path from $x$ to $t$, which you can find recursively.

You can also go from $t$ back to $s$ if you prefer. You can also make progress by finding some point $u$ such that $d(s,t) = d(s,u) + d(u,t)$.