We first need to categorize each interval, assign it a "consonance amount". That's the first problem we find. In the case of the fourth, for example, some consider it perfect consonance, and others consider it a dissonance, depending context (and who you ask).
For simplicity, let's define ours based on Wikipedia's:
1: Perfect consonances: unison, octave, fourth, fifth.
2: Imperfect consonances: major third, minor third, major sixth, minor sixth
3: Dissonances: minor second, major second, tritone, minor seventh, major seventh
1) To what extent does an overtone's first appearance in the overtone series affect the consonance of the 12-TET interval to which it corresponds?
The problem here with "which it corresponds" is that some harmonics will be not very close to its "corresponding" interval. The difference can be as big as 40-50 cent, which is the max difference before the harmonic is closer to another interval. It's more a "middle" than that particular interval at that point.
But let's ignore that. Here is the order in which the intervals appear in the harmonic series, the place in which they appear, and the difference between the interval and the harmonic in cents:
Interval 1st Appearance Difference
root 1 0.0
fifth 3 1.9
major third 5 13.7
minor seventh 7 31.2
major second 9 3.9
tritone 11 48.7
minor sixth 13 40.5
major seventh 15 11.7
minor second 17 5.0
minor third 19 2.5
fourth 21 29.2
major sixth 27 5.9
The first two intervals that appear in the harmonic series are perfect consonances. The third one is an imperfect consonance. So far, so good. But here we reach a stop. The fourth, fifth, and sixth intervals are all dissonant, but they appear fairly early in the harmonic series (harmonic 7, 9, and 12 respectively). Also worth noting that the major sixth, an imperfect consonance, is the last interval to appear.
Other than the first 3 intervals that appear in the harmonic series (root, fifth, and major third), seems that there is no strong relationship. The order is all over the place in relation to their "consonance amount". To me it seems that the first appearance in the overtone series doesn't affect the consonance of the interval (considering the restrictions of the question: interval = closest interval to the harmonic and 12-TET).
As Bradd Szonye mentioned in the comments, it's worth diving into the interval relationship between harmonics. Here, as we go higher in the harmonic series, the interval between harmonics becomes smaller and smaller, quickly converging in one interval, starting with the minor third, as early as the 6th harmonic. So let's see the first 6 harmonics.
4-5 major third
5-6 minor third
Here we see the fourth and minor third. I don't know if "they appear early" has any significance here, given the few harmonics we can analyze before they converge in one interval more and more. What might be significant is the fact that they appear, and all of them are consonant. In fact, if we consider their inversions, those 6 harmonics cover all the perfect and imperfect consonances.
In this case, we do see a strong relationship. Dissonances do not appear in the intervals between harmonics (between n harmonic and n + 1) in the pre-convergence range.
2) If the overtone series is related to consonance as mentioned in #1, to what extent is this consonance reinforced by an overtone which appears repeatedly in the overtone series?
We concluded that they are not strongly related, but let's see the data anyway.
First we need to decide how many harmonics we will consider. If you go deep enough, you'll find that several consecutive harmonics will "belong" (be closer to) the same interval, so maybe it's a good idea to stop before that happens. Also, the higher the harmonic, the lower the amplitude (in general, consider a sawtooth wave for simplicity), so higher harmonics will not have as much as an impact as the first ones.
There is no pretty way to avoid this issue. Tritone appears consecutively in harmonics 22 and 23, and minor sixth appears consecutively in harmonics 25 and 26, but major sixth doesn't appear until harmonic 27, so limiting the analysis to harmonic 22 would exclude it. But that's what we are going to do to avoid consecutive appearances.
Here are the intervals and the frequency in which they appear in the harmonic series, first 22 harmonics:
major third 3.0
minor seventh 2.0
major second 2.0
minor sixth 1.0
major seventh 1.0
minor second 1.0
minor third 1.0
major sixth 0.0
Again, root, fifth, and major third are in the top, and again we see dissonances appearing early. Again, we see consonances in the last places. There doesn't seem to be a strong relationship here either, with the exceptions (again) of root, fifth, and major third.
3) If the overtone series is related to consonance as mentioned in #1, to what extent does the inversion of one of the overtones correspond to consonance?
It shouldn't make a difference in our current categorization. All the inverses correspond to an interval in the same category.
P5 - P5
M3 - m6
m3 - M6
tri - tri
m2 - M7
M2 - m7
4) I have been challenged before when I have said that the higher the overtone is, the less significant it is, because the overtones become less audible as they get higher and higher. Is it true or not true that overtones become less audible as they get higher?
It depends. You can synthesize a sound with any harmonic ratio you can think of. But yes, less exotic sounds will have that tendency, the higher the harmonic the lower the amplitude It's not a constant, this tendency is commonly broken: some higher harmonics might have a higher amplitude than some lower harmonics. Some lower harmonics might not even be present, as some instruments cancel or attenuate specific harmonics.
It's a tendency, not a constant, there are many exceptions. That's maybe why you were challenged about it.
If so, doesn't this mean the lower overtones are of more importance?
The less amplitude an harmonic has, the less it adds to the sound. So, higher amplitude harmonics add more to the sound, they are "more important". As lower frequency harmonics tend to have more amplitude, there will also be a tendency of lower frequency harmonics being "more important", but it's not necessarily the case, as lower frequency harmonics not always have higher amplitude than other higher frequency harmonics.
It would be more accurate to say "higher amplitude harmonics are more important".
Has there been any research done on amplitude of various overtones (say, given a certain instrument such as guitar or piano) and the point at which that amplitude is so low that humans can't hear it?
Interesting question. Don't know of any study in particular, but you can do it yourself.
Use a very steep high pass (low cut) filter and increase the cutoff frequency until you can't perceive a sound. That's where you stopped hearing the harmonic series at that specific amplitude. Now check the cutoff frequency and the fundamental of the sound you filtered. Divide the cutoff frequency by the fundamental and that's roughly how many harmonics you were able to hear.
That doesn't deal with masking and other dynamics, though. Maybe another experiment you can make is the opposite. Use a very steep low pass (high cut) and decrease the cutoff frequency until you perceive that the sound is changing. If you don't perceive a change, it means that those harmonics (if any) aren't adding something to the sound (at least to your ears). When you perceive the smallest change, do the same: divide the cutoff frequency by the fundamental and that's roughly how many harmonics were "important" enough for you to hear a change.
You can then check the amplitude of those harmonics around the cutoff frequency in relation with the fundamental with an harmonic analyzer.
I wonder if the two experiments yield similar results.