Let $(ξ_1,η_1),(ξ_2,η_2),...$ be a sequence of i.i.d.\ copies of a random vector $(ξ,η)$ taking values in $\R^2$, and let $S_n := ξ_1+...+ξ_n$. The sequence $(S_{n-1} + η_n)_{n \geq 1}$ is then called perturbed random walk. We study random quantities defined in terms of the perturbed random walk: $τ(x)$, the first time the perturbed random walk exits the interval $(-\infty,x]$, $N(x)$, the number of visits to the interval $(-\infty,x]$, and $ρ(x)$, the last time the perturbed random walk visits the interval $(-\infty,x]$. We provide criteria for the a.s.\ finiteness and for the finiteness of exponential moments of these quantities. Further, we provide criteria for the finiteness of power moments of $N(x)$ and $ρ(x)$.