Is this possible? Can I leave the airport while on a long transit stop?
Yes, It is possible to leave the airport as long as you have a valid Schengen visa.
How do I state this plan in the cover letter for my Visa application and indicate that I plan to sleep in the airport (not booking a hotel)?
You would definitely need a place to stay (at least for the application). Sleeping at the airport and writing in the cover letter is one option, however, I would suggest you book a hostel or airport hotel and supply that with your application.
Is my baggage going to be through-checked during the layover or do I have to collect it and check it again for the last leg?
It will be checked through and you will get it in New Delhi.