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Anders_H comments on Causal Inference Sequence Part 1: Basic Terminology and the Assumptions of Causal Inference - Less Wrong Discussion
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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..