Who says this doesn't already happen?
Quantum physics actually is the world you are describing. The QM model is based on the fundamental assumption that everything is model able as a wave, known as a quantum waveform. Everything you are looking to have happen actually occurs in QM.
So why don't we find the world we live in looks like a bunch of harmonic waves? It actually does look like a bunch of harmonic waves in areas termed "linear." Motion of objects through air at high speed are often best modeled using waves and harmonics, actually. The issue shows up when something behaves non-linearly.
QM suggests that everything is actually linear, but the "reality" is so complicated that actually modeling the world in this way is too computationally expensive and too hard to measure. If you accounted for the exact state of every proton neturon, etc. in the entire universe, and constructed a quantum waveform for it all, you could actually view the world as a bunch of harmonics!
In reality, we find non-linear models do a better job given our imperfect view of the universe. We like the idea of particles colliding, because it matches well with our billiard-ball style intuition of how things should work.
So you'll probably want to look at what QM does to solve this. The solution of choice is called a Wave Packet. A wave packet acts like a snippit of a wave with some location. It is easy to show that QM lets you break apart the universe's waveform into a bunch of these little wavepackets (with a few minor caveats).
Wave packets are neat because they exhibit "wave/particle duality." In highly linear situations (like a photon traveling through air), the coherent wavelike part of the wave-packets becomes the most driving part, and we see wave like mechanics (which we do see in photons traveling through air). In highly nonlinear situations (like when you use the photoelectric effect to generate said photon to fly through the air), the importance of the wavelike behaviors gets downplayed because there's too many non-linear effects. The importance of position begins to drive, and we see particle like mechanics (which we do see in photons emitted in this way).
A QM lesson in a nutshell, the famous "single photon double slit experiment" is so interesting because the creation of the light is best modeled as a photon (a particle), but its behavior in the rest of the setup is best modeled as a wave (a lightbeam), so both models break down and give bad predictions. Only the full QM waveform model is strong enough to properly predict the uncanny results of that experiment!
So what do we expect in a world based on harmonics? This says that a "consistent" world with an emphasis on harmonics will avoid non-linearity. You won't see a hard wall (like the echo-y concrete surfaces of an underground tunnel). You'll see soft walls which seek to minimize the non-lineary of the situation (nuclear submarines are coated with a rubbery surface whose job is to minimize the nonlinearity between the water and their hull, so they make fewer echos). You would not find people discussing the truthood or falsehood of ideas, but rather find them discussing the harmony of ideas, and balance.
So, given these general principles that lead to a world dominated by harmonics, what might your particular questions yield:
Issue 1: Vibration Q. If the impact of sound causes vibrations that in
turn can cause sound, how might you limit the ripple effects to
maintain a functional world?
The concept of "impact" you are looking for implies a non-linearity. Accordingly, you would see more of those submarine style surfaces which damp the ripples.
Issue 2: Scale Q. If a drum can influence the world, surely a volcano
or a thunder storm are even more exciting than normal. Controlling for
scale seems important, but how to codify such a control?
There are two scales in mind: size and loudness. Loudness would be a simple issue to control: louder things do more. Your intuition for soundwaves would be sufficient for this.
The issue of drums size vs. a volcano's size gets interesting, because theoretically it only matters in one place: its hard to generate high harmonics with a large object, and hard to generate low harmonics with a small object. You would expect to see different "flavors" of effects.
What makes drum/volcano interesting to me is that it starts rapping on the edges of the wave/particle duality issue. If the world is perfectly linear, its size matters little. However, such worlds are not very realistic nor fun for readers. In a world which is simply more linear than ours, emphasizing harmonics more, there are still some nonlinearities. This leads to something which has been found important in many worldviews: a drum usually is drown out by a volcano, but a drum struck with the right feeling and in the perfectly right place can shift the world. I would expect that, in a harmonious world, there are fewer places which such nonlinearity can be exploited, and they have less of an effect. However, they would exist, or else drums would be a far less interesting part of your story.
As a clear example: if you beat a drum and nobody is around to hear it, does it have a large effect? Now do the same beating of the drum in a crowd, it has a much larger effect. Clearly the presence of people causes these non-linearities to become important.
Issue 3: Animals Q. Many animals make appealing noises and sounds, and
it can be assumed in such a world that the number of animals who adapt
to exploit the harmonic nature of the world would increase. What
impact might that have on a broader sense? I'm envisioning a murder of
crows being a truly frightening sight.
This could actually go either way, depending on how you want to sell your story. In one direction, you could argue that the sound of crows is very discordant, so its effects would be more poignant in a world that is highly reliant on harmony. In the other direction, you could argue that these crows exist in this new harmonious world, not our world, and the crows would probably have a commensurately harmonious sound (generating the same feeling of cacophony we feel in our world by emitting a more harmonious sound received by ears that demand more harmony).