Does changing the duty cycle of the 555 timer affect the voltage output like in boost converter? Or does it not change anything except the charging rate of the capacitor?
No, typically the duty cycle would not significantly affect the output voltage.
If you limit the duty cycle to a point where the capacitor can't fully charge/discharge to due on resistance of 555's output pin and diodes, you would see the output voltage sag low.
However, there is no duty cycle that will result in an output over 2x the input voltage.