While hearing different types of equipment gives you an impression, I've got to say until you change very small parts of a piece of equipment, you're very limited in gaining in-depth knowledge. Swapping amplifiers or speakers changes dozens of variables at a time.
Vast majority of engineering work has nothing whatsoever to do with advanced math.
Agreed. People might be rather surprised. When it comes to audio engineering it's rare that you need more than some algebra. There's calculus that explains a lot of what's going on, but it's essentially useless because you already know what it's going to define, and checking with a measurement tool is more valuable than trying to validate your best wishes with the math. Now these are not statements that engineering audio electronics is easy, as it requires more knowledge that's uncommon and unwritten in text books than any other field I can think of that isn't avant-garde/new. And the audio world isn't exhausted yet.
I thought the elephant in the room you were going to mention was sensibility. The ability to keep an unbiased mind, not get emotionally invested with gear, either the one owned by oneself or that one is propagating on the forums - ego and sunk costs tend to keep people going down the blackhole. I think a lot of people hear similar things if they have similar experiences, but the decisions made differ even with the same experiences - and in many cases decisions are not made based on experiences. Of course a live concert background as well as gear experience is required, but if one stays sensible, humble, does not get emotional and ignores the marketing noise (which includes price tags), he can go a long way.
That's a tough one. I've heard systems that I thought were not that good at _____ but the person showing me thought they were, purely based on the assumption that _____ was true because it's easy to believe it. Most of everything I hear people describe about a stereo has almost nothing to do with how it actually works. They're not wrong that they're hearing it, they just don't know what and how the sound is made that they're describing. There's often even a consistency in gear supporting what they're saying, but it's still anecdotal and not the fact of the matter. All in all I think ignoring price is the best possible option, just listen and decide that way what you like. But people form ideas about what equipment will sound like because of experience with someone else that to them appears similar but may not be at all. Pigeon-holing is a problem. Most of it comes from non-understandings by engineers that make products incorrectly - IMO, subpar might be the proper term - and it breeds false knowledge that becomes "truth" to people.