In fact, this is part of why a piano sounds like a piano and guitar sounds like a guitar.
For any piano and especially the upright, the bass strings are actually too short to produce any vibration of the main frequency, the only thing you are left with are the overtones and our brains fill in the rest.
And that brain fill is actually happening across the entire range of the instrument, our brain latches on to specific overtones depending on interval, and the piano tuner (electronic or human), must compensate for inharmonicity in that range.
This means the bass must be tuned lower than the middle which in turn is tuned lower than the upper regions.
The problem with shorter strings is rather simple. Shorter strings are stiffer than their longer equivalents. These strings produce sharp overtones, since there is less length of the string to vibrate on higher frequencies. Perfect overtones need perfect flexibility to be an exact multiple of the main frequency, which is physically impossible.
There is a good explanation on Piano Acoustics linked from the wikipedia page you posted 
Not sure what you're talking about here. For example, the lowest guitar string is an E2 which is 82.4Hz. If you calculate the wavelength, it is 13.7 ft. The string lengths from bridge to nut are only ~25 inches long and yet it's reproducing the fundamental of 82.4Hz without requiring the brain to fill in the gap.
I think that misunderstanding is similar to believing that a speaker can't reproduce 82.4Hz because the cone is not 13.7 feet in diameter, or you can't hear an E2 in a small room because the width and height of the walls is less than 13 feet. If this were true, when you listen on earphones, all 88 keys of a piano and entire range of guitar and vocals would be "audio illusions" of the fundamental frequencies since the physical sizes of the transducers and ear cavity are all less than 1 inch. The highest 88th key on piano is a C8 with a wavlength of 3.2 inches.
My understanding is that it is a bit more complex than that, literally!
The the final waveform is a not just
W = sum(a_i * f_i)
where a_i is the amplitude and f_i are the fundamental frequencies.
It is actually
W = sum(a_i * f_i + sqrt(-1) * (b_i * f_i))
= Psi + i * Phi
Of course, the brain fills up a lot of stuff that is still a mystery but the elec keyboards can set the a_i, b_i to change from "guitar" to "reed organ".
It is the main reason why modelling physically is the best way for realistic results right now - lossy lumped finite element models typically - digital waveguides are one of such models.
In such a model you can incorporate nonlinear damping and resonance functions over time at desired accuracy.
However, I have a good grasp of why instruments sound like they do, so I'm hoping that your statement is a complex way (no pun intended) of showing that the waveform has many harmonics, and that those harmonics vary over time? Not looking for any kind of argument, just hoping for a bit of explanation of the above; From what I've learned over the years it's the balance of harmonics and the way that they change over time that gives an instrument its timbre and explains the difference in tone between instruments despite them playing nominally the same note (i.e. fundamental at the same frequency).
Has to do with the thickness of the core windings, I think? Something to do with the string acting partly as a rigid rod, not just an idealized wire. I hear this also in electric bass strings with super-heavy core wires.
This tutorial series is also illuminating but it's almost too detailed
You might also be interested in AudioKit https://github.com/audiokit/AudioKit, a (macOS|iOS|tvOS) framework for audio synthesis and processing.
 sound design is such an interesting field, as it's both vary artistic but also extremely math/physics/cs/stats if you want.
Can you build vst/au with audiokit for integrating with your daw?
Audiokit is audiounit only but yes, yes it does.
There are tons of tricks employed in guitar playing to create interesting sounds by manipulating harmonics, by both the guitarist and the effect and amplifier signal chain.
Most guitarists that use these tricks are completely unaware of the physical phenomena involved. And the non-guitarist physics geeks always enjoy when I give a short demo with lots of distortion and artificial and natural harmonics tricks.
Recently I’ve been trying to figure out what is special about mridangam and was wondering if I needed to do some analysis myself. Fortunately, I happened to run into CV Raman’s papers analyzing the physics/acoustics/wave forms of a mridangam that are well worth a read.
He first wrote his short paper in 1920 in Nature (almost 100 years ago)
his fundamental thesis/analysis is that the way the mridangam is built is special in that produces harmonic tones (integral multiples of frequencies) which is highly unusual for drums therefore giving it the ability to sound uniquely special, accompany vocal well and to be played in a smaller, softer setting.
A good blog delving into all of this including some cool youtube videos at the end on wave spectroscopy demos using talcum powder (related to raman) are at https://croor.wordpress.com/2010/11/10/cv-raman-on-drums/
A longer version of his paper from proceedings of IIS published in 1934 is here:
Figured some of you would be interested in this.
I've wondered about what similar results hold for Idakka , a drum played in Kerala, and the talking drums of Western Africa .