ble if you use your thumb at the 12th fret) then that E will cause the following resonances in the other strings, n = 4 on the low E string, n = 3 of the A string, nothing noticeable on the other strings even though E might be close to a harmonic for some. The presence of these extra notes will add to the volume of the note plucked. As for sustain, you might think that since these are all the same frequency that they would all suffer the same damping. This is true. But you are judging the "decay" of the note by whether you hear it or not and the added amplitude means that the sound will not drop below detection threshold for a longer time. In contrast if the low E string is excited the same way it will NOT cause sympathetic resonance in the other strings. It will be less audible than its higher pitch counterpart.
This brings us to another point. If you are using your ear to make this judgement I don't trust any of it. The human ear is highly non linear in both amplitude and frequency. Our ears create harmonics from the input. This means that even if the higher harmonics are NOT present in sound YOUR EAR WILL HEAR THEM. There is no way that the physics of the instrument can change this. The ear+brain system hears higher frequencies better than lower frequencies to some extent, possibly related to the last point. Bass and treble notes played at the same driving force will be judged as having different volume by listeners. For a Bass note at 100Hz and a high note at 2000Hz both played pianissimo the Bass note may not be heard by anyone. So any claim made about hearing low pitch notes for a longer time is suspicious without more information.
I can say that on the guitar it is simply not true that the higher pitch notes die quicker than the lower pitch notes. Of course there are too many variables to make any answer to this question complete and absolute. If you are really interested in the behavior of the musical instrument and your own ear for that matter, each variable needs to be isolated and the cause and effect relationship to other variables quantified before trying to make blanket statements about "the instrument". I'd suggest looking at a text like "Physics and the Sound of Music" by Rigden or something non mathematical (assuming you are a musician and not a scientist/engineer/etc) by Fletcher and Rossing.
As a final note I will say this. The hammer placement on a piano means that you will likely excite higher pitch harmonics with each note. This is the opposite situation as my guitar example where I imaging thumbing it on center (like Wes Montgomery). In such cases the lower strings will have a chance to excite many more other strings in the harp, each at the higher harmonic. Again using the guitar example, if I play the low E string but pick it near the bridge I will excite the open string B string (n = 3) and the open high E (n = 4). These will vibrate in their fundamental mode of vibration as those frequencies match the higher harmonics of the low E. NOTE: plucking near the bridge is critical for this to work well. So it is possible that lower stings in the piano have several octaves of strings helping to support the harmonics. But again, as the string motion dies I question if it is the low pitch fundamental that you hear or the ringing of all the harmonics. It would be natural to associate this ringing with the string you hit but that isn't necessarily true. It may be all the others. This in no way contradicts the previous example but serves to illustrate the complexity of the instrument and that with the correct conditions either phenomenon can be observed.
This is a really interesting and complicated question in the physical simulation of string dynamics.
Actually, it’s not entirely true that high pitch notes run shorter. There is a tendency for higher order partials (inharmonic overtones) to decay faster (run shorter). But due to the complexities of piano tuning and string coupling, it is not true that if you play each note on a piano one after another, the next higher note will always decay faster. Also, yo