The rule that's used in the mechanics of rotation is always the right hand rule and that's important. It represents a choice made long ago about how to describe an object's rotation. You know that because if you curl the fingers of your right hand in the direction that the minute hand is turning, your extended thumb will point into the clock. There is no left hand rule because that was not the choice made long ago. Angular speed is the measure of how quickly an object is turning.
Angular velocity is a combination of angular speed and the direction of the rotation.
The downward direction reflects the fact that the minute hand pivots about a vertical axis and that your right hand thumb would point downward if you were to curl your fingers in the direction of the minute hand's rotation. Because there is no torque on it, it isn't undergoing any angular acceleration and its angular velocity is constant. If it happens to be horizontal and motionless, then it will stay that way. But it could also be tilted or even rotating at a steady rate. No, moment of inertia embodies both mass and its distribution about the axis of rotation.