Here's an attempt to translate your problem into a language that is more intuitive for graph theorists:
Let $\Gamma$ be a directed random graph (with loops) of order $n$, where each arc is chosen independently with a fixed probability $p$. Let $\Delta$ be the line graph of $\Gamma$. Let $x$ be the maximal length of a trail in $\Gamma$ (a path in the line graph $\Delta$. \Delta$ of $\Gamma$).
- How is $x$ distributed?
- Under what conditions does $\Delta$ contain a Hamiltonian path?