Riding no hands will depend very much on the geometry of your bike and in particular, the fork. Not all bikes will enable you to ride no-hands unless at very high speed going downhill.

Anyone who wants to improve their circles -- do the one-foot pedalling exercise.

The thing I have found about fixed gear is that it teaches you to keep your hips level and your upper torso from flopping about. This means you are concentrating most on using your legs to move the pedals.