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 as it is one of the prerequisites:
Documents indicating that the applicant possesses sufficient means of subsistence both for the duration of the intended stay and for the return to his country of origin or residence
You need to fulfil the requirements to get the visa and one of the requirement (as mentioned above) is that you should be able to pay for your entire trip and that includes accommodation.
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.