(1) If I have a favorite cinematographer, I study EVERYTHING I can find about them -- their work, of course, but also biographies, interviews, articles, any BTS videos... and if they talk about influential movies and cinematographers, I look at what influenced them. And when they talk about a particular scene, I rewatch that scene. I sometimes pull frames off of a DVD and/or blu-ray just to study the frame in more detail later. I also try to learn about the filmmaking tools available to them at the time, and general popular styles, to put it in context.
(2) I think if you have a favorite cinematographer, or a favorite work of cinematography, it's because something about the image appeals to you so part of the process is understanding and recognizing your own tastes. You don't have to figure out what to look at, you start with what excites you from the start. (Now of course there is another tract, which is when you are studying something like "how to light daytime interiors" or "how to light the woods at night" where perhaps you are analyzing multiple movies by different cinematographers.) So perhaps what appeals to you is the framing and use of lenses of a particular shot, or the lighting effect, or maybe something as simple as how the daytime window glows in the background. The HOW of it all matters less in many ways because there are many ways to approach something -- what matters more is the WHY of it all and having the IDEA in the first place. The visual concepts matter the most because they then lead you to the necessary technique (and techniques change over time so you have to operate from a base that is outside of technology).
(3) I used to take notes when I was a film student but I didn't keep that up, most of my notes had information like you'd see in a magazine article (stocks, filters, etc.) I did use to print off or Xerox a lot of articles that I ran across and keep a file but today that is less necessary because it's online or you have the original in possession. A lot of it came down to memory and re-reading things, repetition. I still remember big chunks of a Gordon Willis interview with an AFI class about "Godfather Part II" in American Cinematographer. I figured if it were important to me, I'd remember it so I didn't need to take notes.
(4) I read a lot, anything I can find in print about the subject. I can now also search discussion groups because often there is some other expert or amateur historian or person who actually has first-hand knowledge on the topic. And I can ask people online or people I know if I have a question. Sometimes you can figure out something, if there seems to be no information available, by studying another movie made by the same person around the same time, hoping that they repeated a technique and mentioned it in an article, or by studying another movie made around the same time in similar circumstances but by another cinematographer.
(5) The range of questions is very wide, depends on what I can't figure out on my own.