Well, like I said, most of my learning has been either in uni, via lectures and course specific notes or maybe the occasional text book. Or by reading online resources usually targetting people with a basic understanding of mathematics (beyond mere numeracy and arithmetic, but still within the capabilities of advanced high school math in most cases, or undergrad level otherwise) If anything, I'd suggest you looks for books on the history of cartography. Introductory Geography text books, preferably used so you don't spend a ridiculous amount, may be a good idea if you can track them down.
Well, that's just the most direct way to understand it. You can still do it right without a mathematical understanding, it's just easier to do with such an understanding, at least in my admittedly somewhat biased opinion as a bit of a math geek.
That depends on how dense the graticules you are using are, how good your eye is, how steady your hand is, and how careful you are. It's like placing a grid over an image, and another grid on a blank page, and then drawing the image on the blank page using the grids as a guide, except that the two grids are distorted compared to one another. Depending on the context, that very imprecision may be something you want. Early cartographers didn't have access to modern surveying and drafting capabilities so they had to freehand it over grids like this so if you want to mimic them, you probably don't want to be too precise.
I'm not entirely sure what you are asking here.
If you want to try this method, you would draw things in on the world map, keeping in mind how the projection distorts things. The Mercator template makes things bigger at the higher latitudes, and the Stereographic one makes things bigger toward the edges. Then, when you want to draw a map in another projection, you would lay a graticule for that projection over your canvas, and draw the features from the original map using the two different graticules as guides. If a coastline crosses 120° W at about 45° N, on the first map, then you would draw the corresponding coastline crossing 120° W at about 45° N on the new map.
That's your call. You haven't posted it so I don't really know anything about it. Even having looked at it, the best I could do is point things that might be relevant to your making the decision.