The SuperCDMS experiment is designed to directly detect weakly interacting massive particles (WIMPs) that may constitute the dark matter in our Galaxy. During its operation at the Soudan Underground Laboratory, germanium detectors were run in the CDMSlite mode to gather data sets with sensitivity specifically for WIMPs with masses ${<}$10 GeV/$c^2$. In this mode, a higher detector-bias voltage is applied to amplify the phonon signals produced by drifting charges. This paper presents studies of the experimental noise and its effect on the achievable energy threshold, which is demonstrated to be as low as 56 eV$_{text{ee}}$ (electron equivalent energy). The detector-biasing configuration is described in detail, with analysis corrections for voltage variations to the level of a few percent. Detailed studies of the electric-field geometry, and the resulting successful development of a fiducial parameter, eliminate poorly measured events, yielding an energy resolution ranging from ${sim}$9 eV$_{text{ee}}$ at 0 keV to 101 eV$_{text{ee}}$ at ${sim}$10 eV$_{text{ee}}$. New results are derived for astrophysical uncertainties relevant to the WIMP-search limits, specifically examining how they are affected by variations in the most probable WIMP velocity and the Galactic escape velocity. These variations become more important for WIMP masses below 10 GeV/$c^2$. Finally, new limits on spin-dependent low-mass WIMP-nucleon interactions are derived, with new parameter space excluded for WIMP masses $lesssim$3 GeV/$c^2$
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