Well, the aspect ratio for that one varies according to the parameters you choose, you can squash and stretch it. The Lambert cylindrical equal-area is just one parameter choice.
Yes; I like the un-squashed Lambert cylindrical precisely because the distortion is intuitive: the equator is undistorted, and everything off the equator has exactly the distortion due to perspective (as viewed from theoretically infinite distance at the equatorial plane). Other vertical perspective magnifications don't have any obvious reason for the choice of magnification, other than "make the map undistorted at latitude X."
I used to write code for these projections as part of my job. Decent choice though.
Mercator's most useful property is you can pick an origin and destination, draw a line connecting, and that gives you an initial bearing for travelling between. Keep that bearing, and you will get there albeit not by the shortest distance. Very handy for sailing ships.
Indeed, each of the projections used has one or another advantage. Mercator's great strength is that it locally preserved directions: a compass bearing of X maps to an angle on the map of X, which, as you point out, means you can plot constant-heading trajectories, which is reasonably efficient if your path is short compared to the Earth's radius. As a consequence, for any infinitesimal area, the map is un distorted. It's globally distorted... but not locally distorted.
I quite like the Winkel Tripel but the inverse is nasty to calculate.
Ah, the compromise solution. In real life, the best solution often is a compromise between solutions that are each bad in different ways.
But since we're talking schools, they'd also be well served by a nice spinning globe.
Indeed: the best map of a sphere is a sphere.