  Let $G=(V(G),E(G))$ be a graph. A function $f:V(G)\rightarrow
\mathbb{P}(\{1,2\})$ is a $2$-rainbow dominating function if for every vertex
$v$ with $f(v)=\emptyset$, $f\big{(}N(v)\big{)}=\{1,2\}$. An outer-independent
$2$-rainbow dominating function (OI$2$RD function) of $G$ is a $2$-rainbow
dominating function $f$ for which the set of all $v\in V(G)$ with
$f(v)=\emptyset$ is independent. The outer independent $2$-rainbow domination
number (OI$2$RD number) $\gamma_{oir2}(G)$ is the minimum weight of an OI$2$RD
function of $G$.
  In this paper, we first prove that $n/2$ is a lower bound on the OI$2$RD
number of a connected claw-free graph of order $n$ and characterize all such
graphs for which the equality holds, solving an open problem given in an
earlier paper. In addition, a study of this parameter for some graph products
is carried out. In particular, we give a closed (resp. an exact) formula for
the OI$2$RD number of rooted (resp. corona) product graphs and prove upper
bounds on this parameter for the Cartesian product and direct product of two
graphs.
