NOR Gate – How Can Output Be ‘1’ if Both Inputs Are ‘0’ Without Power Supply?

Your mistake comes in your logic circuit schema. You have written s instead of c and vice versa. It should be as shown below:

Reset pin going high causes the output to go to zero.

Set pin going high causes the output to go to one.

This is the function of an SR(Set-Reset)-Flip Flop, which acts as a single bit "memory". They latch their outputs due to the interconnected gates, as you see in the first diagram.

Nothing happens to the output when the inputs are not changed. Bad/strange things happen when both inputs are changed at the same time to a LOW state. Circuits designed with these can have strange effects if careful measures are not taken to avoid race conditions or clock(if clocked SR flip flops)/gate delays causing the simultaneous inputs of 0 into S and R.

The truth table of the NOR gate is important because it shows how the two parts of the SR Flip Flop interact - the NOR gate's outputs are fed into each other's inputs, which gives you the latching effect of the output.

You can put both S and R inputs HIGH at the same time if you wanted, but it does not form for to the digital theory of "Q and NOT Q" outputs, so it's not normally acceptable and is called "illegal" in the truth tables.

Bad things happen with both inputs are set low, if both inputs were previously high, because of the gate delays of the NOR gates.This can cause oscillations of the output due to the feedback in the circuit.

You can read more about these race conditions from here and here