How fast do power plants need to respond to demand?
The "fast" response you're thinking of is called frequency response. The grid consists mostly of generators and motors which "electrically" rotate in sync with the grid (the physical rotation being different by the slip). When the load on the grid exceeds the supply, they slow down and frequency decreases (slightly). When the power input to the generator is increased, or the power output of the load decreased, they will attempt to speed up the frequency will increase (slightly). The excitation current of the generator also affects this.

So the inertia of the grid provides the fastest response, where any sudden increase in load or decrease in generation is matched by a decrease in frequency. Then fast-responding plants will increase throttle/power input to the grid and frequency will stabilize. With thermal power plants opening the throttle like this does not change the thermal output of the plant, so if sustained would cause a decrease of pressure at the turbine inlet and loss of power.