How Do You Actually Do Your Mathematics? Better yet, what I'm asking is how do you actually write your mathematics?  
I think I need to give brief background: Through most of my childhood, I'd considered myself pretty good at math, up through the high school level. I easily followed mathematical concepts introduced in my classes and even did a few competitions; I definitely wouldn't say I was a star of the caliber one meets when one ventures out into the bigger ponds, but I thought I was decent and convinced myself I would major in math when I entered college.
That changed after a couple of years when I hit my first Real Analysis class that used Rudin's book; that was the first class, I think, I took that really required more than "expand-a-definition" type proofs and my struggle to find intuition and understanding there impacted my mathematical self-confidence.   I eventually switched majors, with a bit of regret.
One thing that got me, I think, was the veritable explosion of superscripts and subscripts that one encounters for the first time in Real Analysis.  I'd often find myself struggling to set up the machinery of what I was trying to prove, lost in the notation.  How do good mathematicians format their work on paper so as not to get lost in the $i$s, $j$s, and $k$s and keep track of what they're investigating?  I remember dealing with subsequences of sequences to show that limits did or did not exist got especially hairy in this way...writing things like $s_{n_{k_{\epsilon}}}$ and remembering what my goal at each "level" was difficult.  I'd be interested in knowing if aspiring mathematicians and/or professional mathematicians scribble marginalia or have a system to overcome such problems.
Another thing that got me were what I personally called "consider..." statements.  Many times, on this site, the most talented commenters will say "Consider $f(n)$" or "Consider transformation $T:$ $U \rightarrow V$" that in the first case gives a summation that wonderfully telescopes/has an obvious bound, or in the second case transforms the problem into a trivial application of the rank-nullity theorem, or something like that.  Mathematics is a subject replete with geniuses , I understand that, but how do mere mortals investigate such functions and "massage" them into doing what they want?  When good mathematicians get intuitionistic ideas, what (explicit) steps do they take to formalize them, especially when it is likely that first idea is murky or wrong? (Aside: I've been given "use numerical examples" as advice before, but sometimes I think to myself, "I've been dealing with $\mathbb{Z}$ since I was 6 years old, and not so much with Dedekind's definition of the real numbers...") 
There's lots more I could ask, but I want to keep this question tractable, so I guess I might summarize by asking: How do you [professional and aspiring mathematicians] organize your math "notebook", and what perhaps idiosyncratic methods do you employ to be original and clever within it? I know there will be no strict formulas anyone can give; mathematicians are scientists of the abstract; I understand that the subject is acclaimed partly because it's so intellectually and individually demanding.   But I think even acclaimed scientists draw on Springer's Protocols and Nature Methods...There seems to me a bit of a jump between the dryly algorithmic way one is taught to do math in high school and the more abstruse methods at the undergraduate level.  I'd be interested if anyone here could help me bridge that gap, if only for my personal fulfillment.
(Apologies in advance if the question is ill-posed or too subjective in its current form to meet the requirements of the FAQ; I'd certainly appreciate any suggestions for its modification if need be.)
 A: I think these might be actually three different questions. I'll give a brief answer that touches all of them, however.
A good thing to do is: when you have the feeling that you have found a solution to a problem, write it down formally (preferably with latex). This sounds self-evident but it is not a trivial step. Often you find out that your solution is incorrect or incomplete only this way. Also it is obviously a good way to keep your notes organised.
See also: https://mathoverflow.net/questions/1785/how-do-you-keep-your-research-notes-organized
A: There are already several good answers here. I'd like to add a point that hasn't yet been made.  When you are studying mathematics (or almost any subject) in college, you typically follow a path of progression starting with a survey of broad mathematical concepts and methods and, assuming you go to grad school; you end up with a more directed focus on at least a branch of mathematics, if not much more narrowly focused than that.
As you study different areas, you are introduced to different notational methods and symbols. You remember some, but probably don't hold many in memory that you don't use in whatever course you are studying at the time.  It's easy to get overwhelmed. But as you narrow your focus, the notation also narrows and it becomes more important to you because you have invested more time and energy in learning it. And hopefully, because it excites you.  As you become more focused on a particular body of work, the notation becomes second nature and fades into the background while you focus on whatever it is that you are using the notation for.
In my case, my daily work is far from the number theory I studied and I have lost some familiarity with the notation used there.  But I have built my vocabulary in the more applied mathematics I use regularly in my day-job.  In fact, when I read this question, my first thought was that it did not apply to me because the notation I use is so trivial it's obvious to anyone. Then I realized much of it is far out of the mainstream of "general" mathematics and is only obvious because I use it frequently.
To summarize, right now you are wrestling with the notation. But as you get deeper into the mathematics behind it, the notation will sink into the background and you will not see it as a significant challenge.  At least that has been my experience. 
A: I'm struggling with this myself right now! And one of the problems I often have is having a good notation. Normally, I use one notation in my notebooks, but when I make a clean version in Latex I fix them. Once I finish with the latex file, I re-read my proofs, and look for ways to improve the exposition and the notation. I find this a good way to find mistakes, and to make it easier to read. There are always ways to find better notation when you write your proofs in latex and read from the nice version. I find this process of writing your proofs immediately in latex a good practice. I also write dates in my notebooks to know when I did it, and the ideas on that day. Also, I put the name of the latex file corresponding to that part.
But I developed this approach for myself without nobody telling me, and it was based on trial and error. Before that, I gave my advisor a lot of headaches. 
A: In many ways, I am atypical in the way that I approach a problem, but it works for me. Specifically, I try to understand an example in as much detail as I possibly can. If the example, is too complicated, then I make a simpler example. As much of the intricate detail that I can bring to bear on the example is brought. 
For example, instead of trying to understand Lie groups and Lie algebras in general, start with the circle and the line that is tangent at the point (1,0). What is the exponential map? Oh, OK. Now how about $SU(2)$ and $su(2)$? Can you understand that the Lie group is the $3$-dimensional sphere? Can you understand the coordinates? Can you understand the equators? How do $i,j$ and $k$ really work?
What is the difference between the multiplication rule $i\times i =0$ and $i^2=-1$? 
I spend time pondering. And often my notebooks will contain tangential problems or specific computations. I will keep doing the computation until I get it right! If necessary, I will write a program to complete the computation. When I understand the example completely, it is usually easy to abstract. 
Then I follow up, usually writing in a notebook or several notebooks before I begin writing on the computer. I have an advantage in that I have long-distance collaborators, so it becomes necessary to explain the idea to the collaborator(s). That is the first writing stage: write for someone who knows your short-hand and your metaphors. the second stage is to write for someone who does not. Then I write with a set of colleagues in mind, but I assume the colleagues do not remember anything from the previous work. I also try to explicate the notation writing for example "the function $f$, the knot $k$, or the tubular neighborhood $N$. 
A complex analytical colleague only uses $z$ for a complex number, $x$ for a real variable, and $n$ for an integer. These variable choices are culturally determined, and so one keeps with the culture of the discipline unless there is good reason to deviate. As a final example of this, the variable $A$ in the bracket polynomial is known to everyone in the field. The variables $q$, $t$, $X$ etc. are less known and involve different normalizations. So it is the burden of the author to relate these to the more well known choices.
A: Though I left the comment about Gauss saying that "Notions are more important then Notations", I find that when I do mathematics (in my case more of computational and applied mathematics) having good notations/symbols helps me out. Good notations/symbols in some sense makes you "lazy" and if your notations/symbols is good enough, all you might need to do is some "blind" algebra without actually thinking deep. It is like having good variable names when you code up a computer program. You want the variable names to be suggestive and at the same time not too long or messy. I sometimes get too finicky with notations and symbols. If I find that I should have used some other notation or symbol at the beginning, then I actually take pains and the efforts to go back to the beginning and change them throughout. Having good notations/symbols I believe lets your mathematical thought process flow smoothly, especially when you are explaining to others. 
Another habit which I believe is helpful is to always write out your thoughts and ideas. I always face the problem that if I don't write them out, all the thoughts and ideas in my head gets disorganized. If you want to learn a language, you need to speak in the language to learn it. Similarly, if you want to learn Mathematics, you need to think and write mathematics to learn it.
Another thing which I usually do (which sometimes lead others to frown upon me) is to write out the details and pay attention to details, even though it might seem trivial. I understand that this is an infinite process i.e. writing out all the details. Somewhere we need to draw a line, between paying attention to details and keeping track of the overall problem.
Also once every quarter, I take a look at all I have done over the last quarter and "try" to get them in order and organize them. (though I have often failed on this front of organizing them in some nice order).
Another thing I find, especially when I study pure mathematics on my own, is that I am totally lost to find out why certain things are even done or discussed about in the first place. I still remember the day when I did my first rigorous real analysis course when they defined, a closed set as "A set which contains all its limit points". I was like "what?". I was asking myself "What has an innocent looking sequence and its limit got to do with defining a closed set". Where and how did the limits come into picture?
I think this is where a good teacher can help by providing the right motivation for why things are done in a specific way. In fact this is one of the main reasons why I visit this website daily. You have a lot of wonderful Professors and some great students here actively participating and answering questions raised by students like me. Reading some of the questions and answers are really enlightening even though you might have already looked at the problem. Each answer gives you a different perspective of the problem which essentially helps to understand the problem and its intricacies better and helps to understand why things are done in a certain way.
Another important thing to do when learning or doing mathematics is to look for counter examples. I really believe counterexamples play a crucial role in understanding a problem better. Sometimes counter examples help to understand a problem/proof better than the actual proof itself.
A: 
How do good mathematicians format their work on paper so as not to get lost in the is, js, and ks and keep track of what they're investigating?

To be honest, part of this is just getting used to juggling several things in your head at once. Fortunately, as with many other skills, this is trainable: try, for example, doing Sudoku puzzles with a pen.
There are, of course, other ways to keep your work organized. The first thing to do when solving a problem is to write down all of the data you're given and write down the goal of the problem. The second thing is to unravel all of the definitions, working through all the quantifiers. This will get you surprisingly far, at least in real analysis. And then the third thing is to actually think about it. If you can't hold all the quantifiers in your head at once, practice reading the statements aloud slowly until you can (and see the first paragraph).

When good mathematicians get intuitionistic ideas, what (explicit) steps do they take to formalize them, especially when it is likely that first idea is murky or wrong?

Write down an argument that follows the lines of the intuition, at least for a simpler case or version of the problem. If it doesn't work on the simple version, it probably won't work on the hard version. If it does, you can try to figure out whether it extends to the original question and, if not, what the barrier is. 
This is easier said than done. The bottom line is you need to practice the skill of turning your intuitions into proofs. This comes in a few steps: first you need to learn how to prove things, then you need to learn how to train your intuition to help you prove things more easily. Like anything else, this takes hard work and practice and there isn't a magical shortcut. 
Terence Tao has written some very clear stuff on this and related subjects.

How do you [professional and aspiring mathematicians] organize your math "notebook", and what perhaps idiosyncratic methods do you employ to be original and clever within it?

I'm not completely sure what this means.

There seems to me a bit of a jump between the dryly algorithmic way one is taught to do math in high school and the more abstruse methods at the undergraduate level. I'd be interested if anyone here could help me bridge that gap, if only for my personal fulfillment.

Many people I know bridged the gap through competitions such as the AMC. Training for competitions is not everybody's style, but it is a chance to be exposed to interesting topics not covered in the high school curriculum and also a chance to hone problem-solving and proof-writing skills (at the Olympiad level). A book I benefited from enormously while doing this is Engel's Problem-Solving Strategies, which is geared fairly specifically to Olympiad preparation but is a great source of problems and elementary techniques for solving them. For more general advice, Polya's How to Solve It comes highly recommended (although I have not read it myself).
