depends on the instrument. with acoustic instruments the resonance chamber is often tuned so that you can feel the body itself vibrating more when you're in tune
I don't doubt that this is true at some level. But is it really perceptible for 20cent differences? I don't know anything about this, but I'd be surprised. I'd say +/- 20 cents is a bare minimum for calling something "tuned", but low single digits is really the goal.
no, i still use an electronic tuner with my acoustic guitar. but i can feel the body vibrating differently when i get close to being in tune vs being a semitone off