The prediction of plasmonic laser (spaser) and its experimental realization in various systems have been among the highlights in the rapidly developing field of plasmonics during the past decade. First observed in gold nanoparticles (NP) coated by dye-doped dielectric shells spasing action was reported in hybrid plasmonic waveguides, semiconductor quantum dots on metal film, plasmonic nanocavities and nanocavity arrays, metallic NP and nanorods, and recently was studied in graphene-based structures. The small spaser size well below the diffraction limit gives rise to numerous promising applications, e.g., in sensing or medical diagnostics. However, most experimental realizations of spaser-based nanolasers were carried in relatively large systems, while only a handful of experiments reported spasing action in small systems with overall size below 50 nm. In this work, we perform a numerical study of the role of quenching and direct interactions between gain molecules in reaching the lasing threshold for small spherical NP with metal core and dye-doped dielectric shell. We use a semiclassical approach that combines Maxwell-Bloch equations with the Green function formalism to derive the threshold condition in terms of exact system eigenstates, which we find numerically. We show that for a large number of gain molecules needed to satisfy loss compensation condition, the coupling to nonresonant modes plays no significant role. In contrast, the direct dipole-dipole interactions, by causing random shifts in gain molecules' excitation energies, can hinder reaching the lasing threshold in small NP-based spasers.