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.
Full understanding is illusory. If you pursue it, you will find yourself trying to say what a number is, or a set, and digressing into the problem of making language, which for math is a meta-language, precise. And, of course, that can't be done.
So regarding your first question, it might help to observe how futile that innate wish of yours is, and how much you understand without full understanding (or compunction) in all other aspects of your life.
Imagine trying to learn biology and studying the chemical processes in the body, then asking "what is a chemical". You are given an answer that has to do with molecules, a term which you then inspect for precision's sake. Atoms come up, then electrons. Eventually you are learning quantum physics when all you wanted to do was understand how allergens work, or some such thing.
You must operate at the appropriate level for a specific problem. It's no use to reinvent the wheel and do everything from first principles. That would be like writing every program in machine code.
One day, our brains may be augmented with enhancements that allow us to have enough knowledge to understand everything down to our "axioms". Until then, it is a matter of becoming comfortable with our limitations and trying to work with what we have to be awesome.
In terms of knowing which questions are interesting, I think that is one of the harder parts of research. One almost has to be prescient.
And as for getting the important ideas of a proof, my first answer is that sometimes you can't really. Some proofs are just a confluence of numerical estimations and limit results and don't give any real insight into what is going on. Since you seem to be a probabilist, I would point to the proof that a random walk in dimension $n$ is recurrent for $n=1,2$ and transient otherwise. One feels there should be an intuitively understandable reason, but all one gets is Stirling's formula.
For other proofs it is a matter of becoming comfortable enough with the terminology and techniques used in the proof (by re-reading) to see the forest for the trees. In Kung Fu one talks about "learning to forget". You learn the movements carefully so you can perform them without thinking about them when the time comes. You do the same when you learn to integrate or differentiate - you don't want to be doing this from the limit definition when crunch time comes (exam say).