Thank you!

I am not sure your understanding of positivity is right: It is simply a matter of whether there exists any stratum where nobody is treated (or nobody is untreated).

The distinction between "random" and "structural" violations was inherited from a course that did not insist as strongly on distinguishing between statistics and causal inference. I think the necessary assumption for identification is structural positivity, and that "random violation of positivity" is simply an apparent violation of the assumption due to sampling. I will update the text to make this clear.

I'll keep thinking about your question 1. Possibly there are other people who would be better suited to answer it than me..