  We present constraints on extensions to the $\Lambda$CDM 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 \textit{Planck} data.
None of these model extensions significantly relax the tension between
Hubble-constant, $H_0$, constraints from the CMB and from distance-ladder
measurements using Cepheids and supernovae. The addition of the SPT-3G 2018
data to \textit{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 $H_0$ from combined SPT, \textit{Planck}, and
ACT DR4 datasets. While individual experiments see some indications of
different $H_0$ values between the $TT$, $TE$, and $EE$ spectra, the combined
$H_0$ constraints are consistent between the three spectra. For the full
combined datasets, we report $H_0 = 67.49 \pm
0.53\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$, which is the tightest constraint on $H_0$
from CMB power spectra to date and in $4.1\,\sigma$ tension with the most
precise distance-ladder-based measurement of $H_0$. The SPT-3G survey is
planned to continue through at least 2023, with existing maps of combined 2019
and 2020 data already having $\sim3.5\times$ lower noise than the maps used in
this analysis.
