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.