My own personal routine:
I start reading, whenever I feel there is a problem presented in the text (mostly theorems and examples but sometimes questions arise in the text itself), I’ll try solving it. I invest time in this endeavor proportionally to my belief that it can be solved in a reasonable amount of time. If this time runs out, I’ll start reading a little of the solution and trying again. I usually postpone the end-of-chapter exercises to “later” because of spaced repetition and lack of time (I have to have studied a certain amount of chapters for exams so I have to prioritize the quantity.). Sometimes interesting ideas/quasi-problems/ways of looking at things come to mind during the study, and I think on them as well.
When I’m solving a problem, I’ll usually walk around and try to solve it mentally, because I have this uncertain belief that if my math needs paper and pencil, then I can hardly use it “at will.” I’m the kind of person who forgets stuff and needs to repeatedly derive them from first principles when needed, and I can’t quite do that if those derivations rely on writing stuff out. (I also hate memorizing theorems. It feels stupid. :D )
By the way, if the solution is not in the text and I’m stumped, I websearch around. I hate the professors who give grades for the text’s problems and thus disincentiviz solution manuals economically and socially. These take-home exercises correlate more with people’s social skills and network than their effort or knowledge. Quizzing people instead is a much better choice.
1. I am not familiar with any empirical work on the subject.
2. Some textbooks give advice in the beginning, esp. about how ideas are organized. (Which can be helpful if you don't want to read the whole thing - it can explain dependencies between chapters, like 5 requires 1-3, but not 4, etc.)
3. Speculative:
Don't feel like I have enough to contribute to post my own answer but I'm disappointed and slightly surprised this question isn't getting more engagement.