1) From what I have read of reduced relative clauses, I believe so. "Alive" here means "When/while she was alive" and I do believe that it does serve as an adverb in this case, but of that I am not altogether too sure. Other words or phrases like this can also be used, like "Out of office, the President railed against the power of the executive. In office, he decried the executive's lack of power."
2) Having read a synopsis of the text in question, her being a tradition refers to her unique status in the town as someone from the previous generation who must be "humoured and tolerated" according to Wikipedia. It is, as I said, a literary description. As you have, presumably, read all or most of the text, you are in a better place to make a judgement about what it means.
3) The part of the sentence which includes "on the streets without an apron" is enclosed by dashes, so it is not connected to to the part including "remitted her taxes" in the way that I think you are implying. It can be removed from the sentence without making the sentence incomprehensible, and if we do so, we get: "Colonel Sartoria, the mayor, remitted her taxes, the dispensation..." I hope that makes it clearer.
4) a) I think so, although I am not sure. What else it might be I don't know either.
b) Two prepositions are used together fairly commonly in English: away from, out of, up into, up over etc. As I see it, the "on" links with the previous "from...". "Into perpetuity" is actually necessary here, because the dispensation was not created in perpetuity, butbecame perpetual, so it moved into perpetuity.
[Not a teacher]
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