@stepanpruch That's a very interesting point you bring up. I did a great deal of frequency analysis in my graduate study of musicology, and had many of the same questions you do when I started and I can't say all of them have been answered definitively.
The plucked string is a very complicated thing to understand -- as it moves side to side, the string stretches and one might think that would cause a drop in pitch due to it getting longer as it moves off-center. But there's another parameter to consider, which is that as it stretches out, it gets both longer and tighter. The tension of the string counteracts the lengthening, and so it doesn't so much "slow down" as it does "relax". There are subtle drop-offs in pitch with many plucked strings as the sound decays, you can hear it and see it in frequency analysis -- but it's usually in small amounts and easily missed, sometimes due to poor quality instruments or strings, or technical issues such as performer technique or extraneous movements. The point is, the really interesting thing about musical strings is that they can vibrate at the same speed (hertz, or cycles per second) producing the same pitches no matter whether it's loud or soft. When a good string on a good instrument is plucked by a good player, it can speak the same notes (meaning it vibrates and excites the air at the same speeds) in a practically infinite variety of volumes, intensities, timbres, textures, etc. Some smarter people than me will probably discuss this further, but it's a very interesting discussion.
But yes, as you've discovered, pitch isn't as cut and dry as it may seem. Often, especially with the example of a plucked string, and especially during the attack (the first few milliseconds of a sound) the pitch is often skewed considerably, or very indefinite. But we just don't notice because it happens so quickly, or we consider it to be "part of the sound". By the time it's settled into casually ringing out we tend to focus on the nice consistency of a musical vibration, and we consider the blast of frequency clusters that happened at the beginning of the sound to be like a cleansing of the palate, clearing the way for the uniform vibration to start anew.