Time-delayed systems often exhibit multistability of coexisting attractors, which can result in long chaotic transients on the way to one of the coexisting states. Strong enough noise can transform this transient chaos into noise-sustained dynamics. Here we study the interplay between delay-induced multistability, chaotic transients, and noise, in the case of a semiconductor laser with optical feedback from an external reflector. The time-delayed feedback renders the laser multistable, with a set of coexisting fixed points, and induces dynamical events called low-frequency fluctuations (LFFs), consisting of sudden intensity dropouts at irregular times. The deterministic Lang-Kobayashi model shows that, for a large range of realistic laser parameters, the LFFs are just a transient dynamics toward a stable fixed point. Here we analyze the statistical properties of the transient LFF dynamics and investigate the influence of various parameters. We find that realistic values of the noise strength do not affect the average transient time or its distribution, provided the model includes an explicit delay. On the other hand, nonlinear gain saturation has a strong effect: it increases both the duration of the LFF transients and the probability of noise-induced escapes from the stable fixed point. Our results suggest that the LFFs observed experimentally can be, at least in part, sustained by the interplay of noise and various nonlinear effects, which are phenomenologically represented by a gain saturation coefficient.