As you increase collective, you increase angle of attack, and therefore induced drag. So yes, it's safe to say that as you increase collective, you must increase throttle to compensate for the extra induced drag.
In an airplane, as you bring the nose up, you are increasing the amount of lift being produced and start climbing, but as a result you create more drag and lose airspeed. If you want to increase your lift and maintain a constant speed at the same time, you must increase power. Its the same thing in a helicopter, the blades must be at a constant speed, so when you increase angle of attack you must increase power.
Autorotations are a little more complicated. Autorotations work because the air that is rushing vertically through the rotor system creates a very steep angle of relative wind. Lift acts perpendicular to relative wind, so if you imagine the relative wind being completely vertical, the lift is acting on the rotor blades in the same direction as their rotation. This keeps them spinning.
So in the beginning of an autorotation, we lower collective so that we can initiate a descent. The vertical relative wind then begins turning the rotors. Then we have to use the collective to control the amount of lift (and therefore drag) that is being produced. Too little lift, and the rpms will increase and overspeed, too much lift and the rpms will decay and stall.
At the end, we flare because it further increases the angle of attack of the rotor blades. We trade our forward airspeed for this increased lift to arrest our descent.