Is training a neural network on NYT data producing a copy? The answer is clearly no. The neural network is clearly transformative and as a result rights holders have no rights over the transformative work.
The court may try to divide the resulting work into parts based on how much is from the original work, and how much is from the derived work. (See for example the abstraction filtration comparison test). NYT can't claim to own the entire derived work, but they can claim to own the parts that were from the original work.
2. Is the AI spitting out copyrighted material in response to user prompts a copyright issue?
If a copy doesn't exist in some way in the AI's database, then it couldn't reproduce it. Therefore a copy exists (in some way) in the AI's database. Whether it is an interactive conversation or not is irrelevant (ie, there are no exceptions written into the law for interactive conversations).
I will say it is obvious that if a user cuts and pastes the copyrighted text and saves it to a file then they are certainly creating a fixed copy
This is what OpenAI did when they made the training data set.