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I have a couple questions about pre-1900 US stamp identification.
Okay, my comments will be specific to what you are asking about.
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Is there a cross-reference of A Figure numbers to Scott Numbers anywhere?
Yes. In the Scott US Specialized catalog (not sure that is what you meant by "the Scott USA catalog"), there is an "Identifier of Definitive Issues" that lists catalog entries by image reference number. These entries specifically cross reference to the major catalog numbers associated with that image number. When in doubt about an item you are not seeing referenced in the cat, go to the identifier to see if you skipped something.
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I would like to know: does anyone have a process or checklist of examinations and tests they do with these stamps to differentiate them from each other? Or is it pretty much a random process?
Nothing random about it for me. For 19th century it is good to know from memory the face different issues and when they were in use: by year of introduction - 1847, 51, 61, 69, 70, 81, 83, 90, 93 (Columbians), 94/95, 98. Only two sets of commemorative stamps were produced in the 19th century, all the rest are definitives (and therefore appear in the aforementioned "Identifier."
After identifying the general issue then look for technical features in this order:
perforations,
grill embossing,
paper variation,
watermarking
(these all get slotted into the chronology of the visuals). I have listed them in chronological order of first introduction as issued stamps, but as you get to know the chronology of each issue then you will know how to "jump" to the techno points.
This broad outline will help you navigate the catalog more efficiently. But sooner or later you will need to know the issues according to the companies that created them and the kinds of innovation and experimentation they introduced. That is way, way down the road.