We present constraints on extensions to the ΛCDM cosmological model from measurements of the E-mode polarization autopower 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 TT, 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 4.1σ 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.