Dataset copyright is pretty well defined and things can often be used under fair use. Fair use decisions are done with a four prong test and really decided by the courts on a case-by-case basis.
Model weights cannot currently be copyrighted. They are the output of a mechanical process over the dataset. However, software faced a similar situation where the source code could be copyrighted but the compiled binary was not. US copyright law was updated to address this. We may see something similar for model weights.
Model outputs are less clear, but these are likely copyrightable by the user of the model. It is not possible for a non-human to hold copyright, so the model cannot. It is very unlikely that the company producing the model could assert copyright over the outputs. A good analogy here is someone using photo manipulation software.
Super interesting area. I think we will eventually see an update to the copyright code to make weights copyrightable. Also it will be interesting to see how court challenges (code generation, image generation) affect datasets in the future.