How to do philosophy I think you have a grasp of what it takes to make the distinction between philosopher and laymen. To paraphrase, "To look at the original and see the extra-ordinary/unoriginal."
My recent dealings with Aritifically Intelligent Image Generators, kind of concur with what this inquiry gets at. They can abstract or generate images based on their total data-set. We are kind of like that ourselves, plus we have to knit together phenomena/images/symbols to come up with some new abstraction. Making the original unoriginal, perhaps. For more on this I would look to "Paradigm Shifts, Kuhn." I haven't read his book on the matter, too dry for me, but I get the point.
On another point, I think asking and forming questions about certain subjects, fosters both philosophical and creative thought. You can ask questions about any subject, and than rigorously go about finding answers, it is imperative to operate such devices. I have yet to come across anything on scientifically based questioning, but I am sure one could find some data out there.
There are some common methodolgies used in philosophy, such as what you refered to as "axiomatic".What is true of Geometry (Eucilidan Axioms) should also spill over to the philosophies, as it does.