We study the existence, stability, and dynamics of the ground state and nonlinear excitations, in the form of dark solitons, for a quasi-one-dimensional polariton condensate in the presence of nonresonant pumping and nonlinear damping. We find a series of remarkable features that can be directly contrasted to the case of the typically energy-conserving ultracold alkali-atom Bose-Einstein condensates. For some sizable parameter ranges, the nodeless (“ground”) state becomes unstabletoward the formation of stable nonlinear single- or multi-dark-soliton excitations. It is also observed that for suitable parametric choices, the instability of single dark solitons can nucleate multi-dark-soliton states. Also, for other parametric regions, stable asymmetric sawtooth-like solutions exist. These are shown to emerge through a symmetry-breaking bifurcation from bubble-like solutions that we also explore. We also consider the dragging of a defect through the condensate and the interference of two initially separated condensates, both of which are capable of nucleating dark multisoliton dynamical states.