These are all good questions, and not so easy to answer :-) Any answers I can give are personal, of course, and may well have no relevance to your situation.
a): 'Studying' is definitely very important to get an idea of the context of the thing you are trying to solve. (I did a lot of reading both before and during the project). But, it doesn't always relate easily to the problem you have right now. (E.g. I'm using linear scan allocation, and the papers on that method all glossed over the nature of a 'live range', which took me 3 iterations to figure out). And without a practical problem to guide you, it is easy to get lost in literature.
b): What I find is that before I try to do something, I typically put together a high-level overview of how I think I can achieve it (what stuff to change where, and what I will need for that, roughly). But I'd classify that more under planning than under studying per se.
What I also do is take lots of notes. I have several large org-mode files (big fan) and before that paper notebooks in which I write down thoughts as they come up. I think org-mode is superior because it lets me organize things after the fact. Notes are a crucial bit of making progress for me, because I don't need to revisit the same things over and over.
The other thing that I find helps very much (and which I wasn't very good in at first) is to do incremental, small-step improvements. Move a function here, change an struct there, change an interface, move some process earlier or later, split it up. Simple things that you can do and that you know will not go wrong. Many problems can be broken down in that way, and the few that can't can at least be 'isolated' so that you can fix them 'atomically'. I wrote a blog post about this some time ago, if you'll excuse the link: http://brrt-to-the-future.blogspot.com/2016/06/in-praise-of-...
c): I was lucky to be able to join the perl community as part of a GSoC project, and fortunate that the perl community is altogether friendly and welcoming. And I really can't speak for the node.js people at all. But what I will say that anyone with a genuine interest in contributing (as you seem to have) and a willingness to do the work to learn (and to listen) will find a warm welcome from nearly all open source projects, because all of them are relatively understaffed compared to their ambitions :-)
So in your situation, I'd start with checking out the node.js source, start with trying to figure out how a debugger could work (there is lots of prior art here, of course, so you can check out the chromium source as well), and ask your questions basically as the jvns comic demonstrates.
I hope that answers some of your questions.