It has nothing to do with your bottom bracket. It's because of your chain tension, specifically that there is a little bit of slack in your chain.
This is normal and desirable.
Not necessarily. My bottom bracket loosens by itself somehow, even though the lockring is on tight as can be. I always have to loosen the lockring, screw in the fixed cup, and retighten. When its loose its definitely noticeable during trackstands.