For a long time, it has been a complete mystery to me how any of my peers understood any math at all with anything short of filling in every detail, being careful about every set theoretic detail down to the axioms. That's a slight exaggeration, but I certainly did much worse in courses where I attempted to replicate this myself by not reading every proof to save time.
It's only recently, and only within the field of probability theory, that I've developed the ability to do this myself. I am currently following Grimmett's book on Percolation theory. There are far too few details for someone of my level to fill it in completely, but I am getting more than nothing out of it.
Question 1: I would like to learn how to get even more out of such "incomplete" studying.
What tends to happen even now, and more before, is that as soon as I don't understand something, I lose focus and everything just flies over my head. I imagine this is partly psychological, since from a logical perspective if I have to accept proposition $P$ to derive $Q$, I could just think of myself as having proven merely that $P$ implies $Q$, and then no "acceptances" are being made.
Most of the time, professors simply stare blankly at me, wondering how I could persist like this, and all they say is to stop. But it's not that simple, because it appears my intuition is also primarily symbolic. Sure, I think of some geometric pictures when they're called for, but most of my problem-solving creativity comes from pattern matching methods and tricks with situations.
Question 2: How does one distill out the important ideas of a mathematical reading, such as a proof or paper?
Grimmett's book is very helpful in this regard. He will always tell me what's important, and as long as I'm willing to believe him, then I don't have to do anything. But what if I need things that are different than he emphasizes? I always worry that by not understanding everything, I will eventually reach some point in my life when I need to use some fact/method I glossed over and forgot, and that it could be framed in such a way that I would not even be aware of what is missing. That way I wouldn't even be able to do a huge review to rescue the fact from the depths of my ignorance. My current way of thinking about this subproblem of question 2 is that mathematicians always take this risk by not studying everything. So it's a risk-minimization game with time as the constraint. If so, how do I make smart choices with regard to this game?
Question 3: With my recent ability to learn imprecisely in probability, I've started to see many connections, even with outside fields. Many of them are probably fictitious. Many of the questions that I think are highly-motivated might actually be not really worth answering. How does one decide what questions are interesting? As a graduate student who has barely popped out above what the traditional classroom has to offer, I am very lost in this regard.
Question 4: The revelations that enabled me to understand math imprecisely came all at once. A similar comment about the abruptness of my coming upon the ability to proof-check without significant error could be made 2 years ago. Most of my peers seem to learn rather continuously, but the evolutions of my way of thinking seem to come all at once. Is there anything bad or good about this? If so, how do I minimize the bad and maximize the good?
As always, answers to subsets are appreciated.