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Certainly the player's finger are busy. But the string is vibrating a few hundred times a second, so compared to that even a virtuoso's fingers move pretty slowly. Every note, more or less, the player is aiming for Helmholtz motion. When you do a big finger movement or change bow direction there is a transient time while the string sorts itself out, and learning to keep those transients short is one of the reasons for hours of practicing.
As for the "echoing" sound when you stop: there are several ingredients. If you lift the bow off the string, the string can carry on vibrating much as it would in a pizzicato note. If you stop the string you were playing, the OTHER strings may carry on ringing a bit, in "sympathetic vibration". If you stop all the strings, there is a (much shorter) residual vibration in the violin body: the kind of thing you can hear if you tap the violin bridge (carefully) with a pencil. Finally, there may be an actual echo, from the room you are playing in. Some of those things are going on all the time as you play: but the first one would be killed off if you play your next note on the same string.