Quite an impressive review!
Just 2 small correction -
1) There is another compact surface besides the 4 you have mentioned that can be constructed from a flat 2D square. Its constructed by gluing each pair of opposite edges with a 180 degree twist. It's called "Boy's Surface".
2) A 2D Torus T^2 has a only Extrinsic Curvature, but NO Intrinsic Curvature.