My second question was, what happens when the circuit is saturated ? Does it still work ? Why is saturation "bad" ?
Saturation is only bad if it results in distorted output.
If you have a 2 V peak to peak input signal and a gain of 10 the output will be distorted unless the opamp can produce 20 volts peak to peak output. An opamp with + and - 12 volt supplies may be able to do this, but one with + and - 9 volts supplies will not be able to do this.
Saturation is just a normal consequence of the internal construction of the opamp. The opamp can't produce more output than its supply voltage and it usually can't output as much as the supply rails either.
Also, unless the output is exactly centred between the supply rails, one direction of swing will saturate before the other. This might result in "flat topping " of a sinewave, for example and distorted output at less than the saturation limits.
If it happens, you just need less signal, less gain or better adjustment of the DC conditions in the circuit.
I found this confusing:
I mean if I have 15V and I need to power something why can't I just use the 15V instead of using an op-amp only to realize 13V as my output voltage ?
I do not recall anywhere in my post where I stated that " op-amps are used to provide a constant dc voltage".