At some point the waves will have dispersed so much the sound vibrations would be smaller than the random motion of the air molecules. There would be no way to detect the original sound at that stage.

Technically, yes. It’s like ripples in a pond. Sound waves are energy, energy doesn’t just disappear. But it decreases significantly over distance, we’re talking down to 0.00000000…..1% of its initial intensity. Now try and pick that one specific wave out of every other noise that’s ever existed.

Yes, but not at the range of thousands of miles. You can use large (3-4 meter wide) parabolic dishes to focus the sound waves to a point. They're called Acoustic Mirrors they had a range of about 35 kilometers. You can also buy a parabolic microphone, those can pick up sounds up to 2.5 kilometers away.

It is possible to pick up extremely weak signals, even if there's background noise. For example, and this is radio and not sound but it illustrates an interesting point, the Voyager space probes are in interstellar space and their transmitters aren't any stronger now than they were when they launched. However, we're still able to communicate with them.

Point a very sensitive antenna at the sky right near, but not at, where you're looking for the signal; what this antenna will pick up is the background noise. Point another antenna exactly at the Voyager probe; this antenna picks up the background noise and the probe's signal. Subtract the two signals from each other and you're left with just the probe's signal.

So the answer is “yes”. Enter the Bloop. Note here we are talking about ultra low frequency, high amplitude sound that is traveling in the deep ocean.

Similarly, earthquake detection also rely on detecting “sound”. The waves generated by earthquakes are also low frequency and high amplitude. In this case, the sound is traveling through Earth’s mantle, and can be detected by sensors around the world.

But when it comes to the kind of sound that we humans can hear, that travels through the air — the Earth is just too noisy of a place for weak sound signals to be detectable over noise at thousands of miles away.

Inverse square law, double the distance, halve the volume of the sound. Eventually the kinetic energy of the sound, which is moving the particles of air, will have transferred all of its kinetic energy to its surroundings and it will then cease to exist as a sound.