We present constraints on extensions to the ΛCDM cosmological model from measurements of the E-mode polarization auto-power spectrum and the temperature-E-mode cross-power spectrum of the cosmic microwave background (CMB) made using 2018 SPT-3G data. The extensions considered vary the primordial helium abundance, the effective number of relativistic degrees of freedom, the sum of neutrino masses, the relativistic energy density and mass of a sterile neutrino, and the mean spatial curvature. We do not find clear evidence for any of these extensions, from either the SPT-3G 2018 dataset alone or in combination with baryon acoustic oscillation and Planck data. None of these model extensions significantly relax the tension between Hubble-constant, H0, constraints from the CMB and from distance-ladder measurements using Cepheids and supernovae. The addition of the SPT-3G 2018 data to Planck reduces the square-root of the determinants of the parameter covariance matrices by factors of 1.3 − 2.0 across these models, signaling a substantial reduction in the allowed parameter volume. We also explore CMB-based constraints on H0 from combined SPT, Planck , and ACT DR4 datasets. While individual experiments see some indications of different H0 values between the T T , TE, and EE spectra, the combined H0 constraints are consistent between the three spectra. For the full combined datasets, we report H0 = 67.49 ± 0.53 km s −1 Mpc −1 , which is the tightest constraint on H0 from CMB power spectra to date and in 4.1 σ tension with the most precise distance-ladder-based measurement of H0. The SPT-3G survey is planned to continue through at least 2023, with existing maps of combined 2019 and 2020 data already having ∼ 3.5× lower noise than the maps used in this analysis.