Question

2) How do you know that different frequencies of sound waves all travel at (very nearly) the same speed?

Answer 1

The speed of an individual air molecule affects how fast it vibrates between regions of compression and rarefaction, which increases or decreases the pitch of the sound, but the molecule doesn't actually travel far from its original location. When you clap harder, most of the extra energy is spent moving more molecules. Slower hands move less molecules because the instantaneous velocity of each particle is much higher than that of your hand, so particles travelling away from the hand aren't moved by the clap. As the hand goes faster it catches up with the slower particles or the ones moving at angles to it, thus affecting more particles.
When the extra energy is used to speed up the molecules, It increases the pitch of the sound you'll hear, but the speed at which nearby molecules realize the increased pressure and start to flow away from it depends greatly on the overall density, temperature, etc of the air.

I probably should note that due to the increase in pressure and temperature very near to your hands, the speed of sound does technically increase right there. But as soon as the sound wave begins to move away from that area, it almost immediately returns to normal.