A) The purpose of egg crate shape is to disperse sound waves of different frequencies. The in and out curvature of egg-carton-like surfaces irregularly reflects many midrange frequencies from many different points along the wall. Egg carton-like shapes are not designed for soundproofing per se, but they help with very “live” rooms to reduce sound reflections. Acoustic Blankets, especially hung away from the wall a little, actually provide better soundproofing.
Other shapes like commercial sound-absorbing material that looks like little wedges of foam at right angles, can soak up many frequencies pretty evenly. Similarly, bass traps that are essentially large cylinders stuffed with fiberglass insulation or foam that soak up bass energy, especially in corners, also used to even out the sound in a room.
Very little bass energy is actually absorbed by egg cartons, and that’s most often what bothers the neighbors.