There are a lot of factors to consider here. When analyzing the tone that is produced by the attack we usually use a simplified model in which the string is ideal and fixed at the two ends by immovable rigid constraints. We also don't consider changes in length of the string due to stretching as it moves, etc. This is not a bad model for a solid body electric. For classical and other acoustics the bridge will move.
The string model supports an infinite series of harmonics, all related to the fundamental (f0) by n*f0. n = 1 is the fundamental (the note that the string is tuned to) and the higher ones are n = 2, 3, 4, etc. The n = 1 is the 12th fret harmonic and is an octave above f0, n = 3 is an octave and a 5th, etc.
One can attack the string many ways, with a pick, a bow, striking or tapping with a stick (some guitarists will use a drum stick). And one can angle the pick differently to produce different tones. The action of a pick can be hard to model. Some treat it as pulling the string away from equilibrium by a small amount then releasing the string from rest. Others may treat the string as sliding off the pick like a ramp. The initial conditions of the string at the moment of attack (or moment after attack) determine the number of harmonics present in the waveform, both mechanically in the string and acoustically in the sound field produced by the instrument. The modes corresponding to each harmonic are sine curves from trigonometry. These are rounded, sort of semi-circular arcs with "nodes" at the ends, and at even spaces between the ends. The n = 2 harmonic has a node over the 12th fret (the mid point), n = 3 has 2 nodes in between, etc.
Consider the "plucked string" as an initial shape that is suddenly released and allowed to move, even though it is probably an over simplification. If the pick is very small (narrow) shape is roughly a triangle (a model used in some basic physics books to motivate the analysis of tone). To describe the initial shape requires a lot of sine waves all added together. Thus, for any attack you basically produce a whole series of harmonics. The math involved is basic calculus but in a nut shell you can predict the strength (volume) of each harmonic. One constraint is that any harmonic with a node at the location of the pick will not be excited. Another point, though not a mathematical constraint, is that "typically" (but not always), the higher harmonics have smaller amplitudes.
Plucking a string right at the mid point (with the above model) will necessarily KILL all harmonics with a node there including n = 2, 4, 6, 8, etc. All even harmonics are absent. Also, the odd harmonics will be, typically, weak. If you were to use your thumb rather than a pick the initial shape would be less pointed and more rounded, similar to a true sine curve. In this case you would have almost all n = 1 present in the harmonic spectrum of the note. Playing closer to the bridge would necessarily kill some harmonics but is you are really close you may kill the 21st harmonic (which is already weak) and excite a large amplitude n = 2 through 20. The same type of analysis can be applied to the hammered and bowed strings to predict the harmonics present. Even though the models are an over simplification of the attack they do pretty well.
Here's how this relates to tone. When we pluck a string we are never just playing the note the string is tuned to but producing a harmonic spectrum. It turns out that the n = 4, 5, 6 harmonics produce the major triad. So why don't we hear a major chord with every note? A few reasons. One is that these are all way above the fundamental in pitch and very low amplitude. What we hear as the note is the fundamental and the harmonic spectrum is perceived by our brains as quality of tone (or perhaps timbre, though I could be confusing these terms). "Bright" sounding notes have more high pitch harmonics (with noticeable strength) whereas "Warm" tones have fewer of the harmonics present or have them present with much lower amplitude.
These results are affected by the response of the instrument itself, acoustic or electric, presence of effects, etc. Wes Montgomery used his thumb and frequently struck over the finger board (not just near it). Of course his set up was probably already "Warm" electrically speaking. But the use of the thumb over the finger board made it warmer.
One can tweak out different combinations of harmonics at the exact same location on the string by adjusting how you attack the string, as I mentioned. Even with a plucked attack there are many ways of angling the pick or finger that will change the quality of the sound. Some flamenco guitarists move their finger closer to the bridge to facilitate fast tremolo, and this changes tone. I've seen some classical guitarists change right hand position to change tone. Pepe Romero commented that it would be better to adjust the angle of attack and other factors while leaving the hand at the optimal position for best tone. Of course that is all unnecessary on an electric but you will hear the difference if you plug in with a clean set up and start picking in different areas.
With respect to the electric you need to consider pickup placement. The pick up responds to the motion of the string right above it by electromagnetic induction. Any harmonic with a node over the pickup will essentially be killed in the signal sent to the amp. Hence, even if you play near the bridge but have the neck pickup on (and bridge pickup off) the resulting tone will be warmer than if you switch your pickup config.
Lastly I mention that the decay will be frequency dependent. The simplest model that can be solved assumes a linear damping force. The higher the frequency the faster the decay, but this is always true and not really dependent on the attack. What is attack dependent is the initial strength of the harmonics. Eventually all notes will decay to leave the fundamental as the dominant harmonic. It should be mentioned that the ear produces aural harmonics due to non-linearity of the basilar membrane, so it is not likely that any human has ever experienced a pure sine wave.
