How can you with Python 2.7 and Selenium read the POST or GET request that the driver send when the driver clicks on a button that send some request? And is it also possible to read the response?
I don't know whether some libraries will do this for you. But I think you can simply set up a thread to run something like tcpdump to capture all HTTP package and store them somewhere while the test is running on the main process. 
You can start the thread before clicking the buttons and do some analysis on the captured packages after your test to get those packages containing the request you want.
Python Selenium, read POST/GET Request when clicking a button