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If that CUPS thing is on my machine, it's sure doing a good job of hiding. It only shows up if I go to that URL while online.That's very interesting, but it's not responsive to my question, namely, does "that CUPS thing" > Completed Jobs contain the fax records for which you've been looking?
(If you'd like to know more about "that CUPS thing," I suggest that you post a question in the Networking Forum.)
Aside: I know nothing about this app, so I'm not recommending it, but
SimpleKeys for Mac - Assign functions to your keyboard F-keys addresses one of your previously posted needs. (There are similar apps, some free, some more or less costly.)