No, moment of inertia embodies both mass and its distribution about the axis of rotation. The more of the mass that is located far from the axis of rotation, the larger the moment of inertia. For example, a ball of dough is much easier to spin than a disk-shaped pizza, because the latter has its mass far from the axis of rotation.
A balanced seesaw is simply one that isn’t experiencing any torque—the net torque on it is zero. 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.
Angular speed is the measure of how quickly an object is turning. For example, an object that is spinning once each second has an angular speed of “1 rotation-per-second,” or equivalently “360 degrees-per-second.” Angular velocity is a combination of angular speed and the direction of the rotation. For example, a clock lying on its back and facing upward has a minute hand with an angular velocity of “1 rotation-per-hour in the downward direction.” 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.