...for the Madrid - Paris leg you have a lot of options, check all major airlines.
NOTE that if you are flying out of Paris with another airline, make sure you fly in to the same airport... or that you have enough time to change airport.
Spanair and Iberia also have Santiago - Barcelona flights. So it might be an option via Barcelona to Paris as well.
I have also heard of a flight Vigo (one hour drive south of Santiago) - Paris direct flight, I did a search for it but could not find it..... it might be an option as well if you find that flight. They may not fly it daily.