Before I started learning piano, and when I was a beginner, I told myself I would never, ever perform all those gestures and movements. I thought it would be embarassing and always wondered if the swaying and movements were forced or not.
But eventually I noticed I was moving a bit while playing naturally. I don't really move a lot when I play, and it's not dramatic at all, but it's still there. A lot of times my movements helps me keep the pulse of the music, or make the music sound better (I move and hit the keys to produce a certain tone, sometimes I move subconsciously to produce a certain dynamic)... other times I'm just absorbed in my music and feel the emotions.
As for looking away, I'm quite a visual person, and sometimes I look away and imagine imagery that goes along with the music. It helps me play better, and also keeps me from overthinking and messing up the piece, especially during a performance when I'm nervous.
I would feel uncomfortable if my movements were forced/exaggerated though. I have friends whose teachers tell them to make gestures to show more emotion. I don't think I'd be able to do that. I would definitely be embarrassed, and distracted from the music.