The mass is included in the moment of inertia:
I = (3/10)*m*r²

Moment of inertia is similar to "mass" except with rotating objects.

Objects with large mass require larger force to accelerate. Objects with large moment of inertia require larger torque to start them rotating. The reason they're different is because it depends how far the mass is from the axle (or center of rotation). If the mass is far from the axle, like in a bike wheel, then the moment of inertia is fairly large even though the wheel is light. It's the same principle as angular momentum (gyroscopic momentum).

Height doesn't matter, as long as the cone is made from a uniform material, the same density everywhere.

I'm still not sure what you mean by calculating the torque for that system. If the system is rotating at a constant speed, the torque is zero, since it is not changing speeds. Did you mean gyroscopic precession instead?

Work is the amount of total energy you've added to the system.
E = (1/2)*I*u²:
kinetic energy equals moment of inertia times rotational velocity squared.
Rotational velocity is usually given in radians per second.

Here's an example of calculation of work.
If you take a 2kg cone with widest radius 0.1m, the moment of inertia is:
I = (3/10)*m*r²
= (3/10)*(2kg)*(0.1m)²
= 0.006 kg*m².

In order to accelerate this to 300 RPM, the amount of work you would have to add is:
E = (1/2)*I*u²
= (1/2) * (0.006 kg*m²) * (300 rotations/minute)²
= 270 kg * m² * (rotation/minute)²
One rotation is 360 degrees or 2*pi = 2*3.1416 radians. So multiply by (1 minute / 60 seconds)² and (2*3.1416 radians / 1 rotation)²...
E = 270 kg*m² * (rotation/minute)² * (1 minute / 60 seconds)² * (2*3.1416 radians / 1 rotation)²
= 2.96 kg*m²/s²
= 2.96 Joules of work.