2.8 Assumptions for interpretation and statistical inference
I have nothing to add, here.
The brms package is quite general and allows users to fit models from a variety of likelihoods other than the Gaussian. For example, users can accommodate outliers/extreme values with Student’s t regression. You can do count regression with the Poisson or the negative binomial… For more, see McElreath’s lecture introducing the generalized linear model or Bürkner’s vignette, Parameterization of Response Distributions in brms
The brms package can also accommodate homoscedasticity with distributional modeling. In short, one simply models \(\sigma\) in addition to the mean, \(\mu\). See Bürkner’s handy vignette on the topic.
And the issue of independence is where the multilevel model comes on. See any relevant text, such as Statistical Rethinking or Data Analysis Using Regression and Multilevel/Hierarchical Models.