I've seen other questions on this site asking for problem-solving tips and general problem-solving strategies but I haven't seen any questions asking for the thought process of professional mathematicians as they approach problems (although if there are some, feel free to link them). What are the first things that go through your mind when you read a problem? How do you understand what the problem wants from you, or what knowledge the solution requires? How do your methods and your process change as the solution becomes longer or more crafty? Anything else that goes through you mind as you work through a problem. As someone fairly new to rigorous, proof-based math, I am really interested in answers to these questions.
Below, I've provided a few problems from Lang's Linear Algebra I was hoping could be used to illustrate a professional mathematician's thought process.
Chapter 8, section 4, exercise 20:
Chapter 7, section 3, exercise 14:
Chapter 5, section 4, exercise 4:
If it's of any relevance, I am nearing the end of Lang's Linear Algebra and I have previously studied most of Spivak's Calculus. I have largely felt comfortable with the material in both books. Some techniques I have found useful in problem solving include making sure I understand the relevant definitions and specialized terms used in difficult questions question, looking for similar examples in the text, and trying to see if I can recognize any previous methods of proof that may apply to the problem at hand.
I'm not sure this answers your question, but it may be part of an answer to a question behind your question.
As a professional mathematician I hardly ever think about problems, if by problems you mean short statements like those in textbooks asking you to prove or disprove something. I'm much more interested in understanding what is going on that might explain some example I come across, or a similarity I sense between two seemingly different mathematical structures. I'm often working out examples and guessing and then checking conjectures. It's only toward the end of a research project that I can formulate a theorem clearly enough to look for a formal proof - and by that time I am pretty sure I will be able to find one, because all the preliminary work suggests how it ought to go.
As a teacher I do think about problems to set my students. In a narrow sense I want my students to see that they must read carefully, understand the definitions, know how to prove things about what we are studying. But even there I try to find or invent ones that resemble research as much as possible - asking them to examine examples, make conjectures, then prove or disprove them. It's not "real research" only because given what I know that they don't yet, I can formulate an adventure that they are likely to succeed at and learn from.
The only time I encountered problems for their own sake was when I coached students for the Putnam exam. Then the techniques you mention in your question are the starting points.