Do soundwaves heat up the air through which they travel?

Yes, but I bolded 'totally isolate' because in any real experiment you will have heat loss to the walls of the 'room' and also the heat generated by the electrical wiring and the mechanical friction of the speaker head, and these effects will all be much larger by the time the system comes to thermal equilibrium than the contribution of thermal energy to the air from the sound waves.

 

The pressure oscillates so, on average, there is no effect. The only net effect is actually due to the fact that there process isn't exactly isentropic, some energy is dissipated. Of course, all that sound energy must go somewhere. If it cannot escape, it must all be dissipated. So, if you have a loudspeaker producing 100 Watt of power in the form of sound energy and none of that energy leaves the gas, then you must reach some stady state situation in which you end up heating the gas with 100 watt.

 

Yes, because sound waves attenuate in air; i.e., the sound power level drops off faster than 1/r^2. Here are some attenuation numbers at sea level & STP:

20,000 Hz 528 dB per kilometer
2,000 Hz 9.88 dB per kilometer
200 Hz 0.95 dB per kilometer
20 Hz 0.0127 dB per kilometer

Because the air pressure remains the same before and after, the energy heats the air up.

See http://www.csgnetwork.com/atmossndabsorbcalc.html

 

I vaguely recall an elegant (and obscure) derivation of the speed of sound given the specific heat (maybe vice-versa). I can't seem to find the derivation right now, but it seems to be attached to the names Carnot, Laplace, Huginot, and Hadamard.

IIRC, the assumption is that the process is adiabatic.