In my project, I would like to adjust the duty cycle of the Edison board such that it sleeps for a certain number of cycles being idle doing nothing and then wakes up again in a repeated fashion. Is this possible?
It's possible to change the duty cycle of PWM signal and if you want to change its value during a process what you can do is to do system calls via your code so that from time to time the duty cycle behaves as you want. I would also suggest you check this Multiplexing guide for the Edison, http://www.emutexlabs.com/project/215-intel-edison-gpio-pin-multiplexing-guide http://www.emutexlabs.com/project/215-intel-edison-gpio-pin-multiplexing-guide.
Do you know how can I do this using python code? Moreover, what I understand from the guide is that PWM controls certain pins not the board itself or this is not the case?
You are correct, some pins of the board are PWM capable, but this feature will work only in that specific pin.
About examples using Python, I would suggest you to check this https://github.com/MakersTeam/Edison/tree/master/Python-Examples link, there you'll find some basic but useful examples. For PWM usage, you can take a look at the Servo sketch.