Great post, I have often noticed the same thing, that a classical musician can play any piece of music put in front of them like a robot, but can't improvise in the moment, like a blues player, or Jimi Hendrix could. I always associated improvisation as being kind of wild and free with the playing, but as you mention Bach was a master of improvisation and it came out sounding like a well thought out composition because he was so good!
RE: Improvisation 2. What some might say is a "Lost Art" in the classical community, I prefer to say "not trending."