Inertia. A flywheel is meant to have a mass that is, compared to the system it is in, rather large. Once the flywheel is brought up to operating speed, it's inertia helps to eliminate or reduce speed fluctuations in the system.
No, I think the abillity to do work is potential energy. Energy is the sum of kinetic energy and potential energy.
So you are implying that only potential energy can do work? How would you explain a collision situation when an object (such as a billiard) collides with another billiard to change the kinetic energy of the collided billiard(changing it's velocity to a nonzero value, thereby giving it a kinetic energy)? (any change in energy is caused by work being done, therefore the colliding billiard did work on the collided billiard.)