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These are great questions, and their answers can be found in the paper Why the logistic function? A tutorial discussion on probabilities and neural networks by Michael I. Jordan, 1995. I highly recommend you read it all. However, I will provide a short summary here. Note that some of what I write will not necessarily be found in the paper.
In the context of ...
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There is no "golden rule" when it comes to anything related to ML, models and metrics. A common approach is to build a simple classifier
and use it to compare against more advanced models. IMO every single metric should be understood in context. Perhaps "65% accuracy" might seem low, but if the baseline model is offering, say, 55%, that ...
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Sorry that this section proves a challenge! (I corrected the question to indicate that the Uniform on $(a,b)$ is imposing the inequality for all $i$'s, as this is (too) implicit in the book.) There is alas a typo in the book when defining the set (second displayed formula on p.220), not a mistake in your reasoning!
The constraint on $(a,b)$ is thus that for ...
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