It depends - the "v^2" term quickly passes the linear "m" term for some scenarios. If I've got a choice between making it ten times as heavy or capable of spinning 10 times as fast, I don't want more mass there.
Well, in the context of storing energy, you want mass in the flywheel. Having a lightened flywheel is beneficial in, say, racing applications however... But that's not really important in this discussion.
No - as the immediate prior post correctly stated, stored energy increases linearly with m (flywheel mass) and as the square of rotation rate (omega^2) So a flywheel having 0.1m rotor mass spinning at 10omega stores 10 times as much energy as a flywheel of 1.0m spinning at 1omega. 10% of the mass, 1000% of the stored energy of the heavier, slower rotor.
Of course, the rotor has to be able to handle increased circumferential and radial stresses resulting from increased omega. Carbon fiber (like T1000 grade) has extremely high tensile strength (lets you spin a carbon fiber rotor very fast) but low density. T1000 rotors store more energy at lower mass than steel (or any other metal) because of the proportionality to omega^2 and their high tensile strength.
But for commercial storage, it's really about the $/kWhr/mass in terms of the overall rotor economics. By that metric, it's hard to beat a composite rotor with E-glass as the major fiber. It's cheaper than carbon fiber by easily 100X and has about 30% to 50% of CF tensile strength. It's a bit denser than CF, but its cost metric is why most modern large energy storage flywheels use glass/CF hybrid composites and magnetic bearings (for high speed and zero wear).
In the old days, they made flywheels out of solid steel, then switched to piano wire (higher tensile strength). Then fiberglass, aramids and carbon fiber happened.