The rock and rock-ice mixtures of the core-enveloping spherical shells comprising terrestrial body interiors have thermally determined viscosities well described by an Arrhenius dependence. The activation energies characterizing such bodies can give rise to viscosity variations exceeding 10^40. We first explore the impact of a cut-off to limit viscosity magnitude in cold regions. Using a spherical annulus geometry, we investigate the influence of core radius, surface temperature, and convective vigour on stagnant lid formation resulting from the extreme thermally induced viscosity contrasts. We demonstrate that the cut-off viscosity must be increased with decreasing curvature factor in order to obtain physically valid solutions. We find that for statistically-steady systems, the mean temperature decreases with core size, and that a viscosity contrast of at least 10^7 is required for stagnant lid formation in smaller core systems. Inverting the results from over 80 calculations featuring stagnant lids (from a total of approximately 180 calculations), we apply an energy balance model for heat flow across the thermal boundary layers and find that the non-dimensionalized temperature in the Approximately Isothermal Layer (AIL) in the convecting layer under a stagnant lid is well predicted by our parametrization. Moreover, the normalized thickness of the stagnant lid can be obtained from a measurement of the non-dimensional surface heat flux once the temperature characterizing the AIL is determined. Stagnant-lid thicknesses increase from 10 to 30 percent of the shell thickness as core size is decreased, and thick lids can overlie vigorously convecting underlying layers in small core bodies, potentially delaying secular cooling and suggesting that small objects with small cores may have developed thick elastic outer shells early in the solar system's history. However, we also find that for the small number of 3-D calculations that we examined, parametrizations based on 2-D calculations overestimate interior temperature and lid thickness in smaller core systems.