I understand that playing a square wave from speakers cannot produce a PERFECTLY sharp division between compression and rarefaction. But it's sharp enough to sound distinctly different from a sine wave. As it travels, does the wave "smoosh out" even further as it travels, changing from a square wave into something closer to a sine wave?
Yes. Higher frequencies are attenuated more over distance than lower frequencies are, which has a rounding effect on the square wave as the upper harmonics are reduced.