  We investigate some topological and spectral properties of
Erd\H{o}s-R\'{e}nyi (ER) random digraphs $D(n,p)$. In terms of topological
properties, our primary focus lies in analyzing the number of non-isolated
vertices $V_x(D)$ as well as two vertex-degree-based topological indices: the
Randi\'c index $R(D)$ and sum-connectivity index $\chi(D)$. First, by
performing a scaling analysis we show that the average degree $\langle k
\rangle$ serves as scaling parameter for the average values of $V_x(D)$, $R(D)$
and $\chi(D)$. Then, we also state expressions relating the number of arcs,
spectral radius, and closed walks of length 2 to $(n,p)$, the parameters of ER
random digraphs. Concerning spectral properties, we compute six different graph
energies on $D(n,p)$. We start by validating $\langle k \rangle$ as the scaling
parameter of the graph energies. Additionally, we reformulate a set of bounds
previously reported in the literature for these energies as a function $(n,p)$.
Finally, we phenomenologically state relations between energies that allow us
to extend previously known bounds.
