Not Using a Segment/Leg of a R-T airline ticket

Its called a hidden city ticket and it is illegal. People have been sued by airlines. Google hidden city ticket. People do it. Depends on your tolerance for risk

On the return, How could I insure that there would be a seat for me
since I would have never “checked in” on the flight in Barcelona?

You can't ensure that the airline will act faithfully by the terms of the ticket when you are contemplating violating the terms on your end. The airline will cancel the rest of your itinerary. Simply put: Don't do it.

If you miss any segment on a multi segment flight, most airlines will simply cancel the remaining segments and will owe you nothing according to their contract of carriage.

So, if you book the round trip to Barcelona and then get off the plane in Madrid and go no further on that ticket, you will have to purchase a new ticket from Madrid back home.

As Jessica notes, you are looking at hidden city ticketing, a violation of the airline carriage contract with the applicable risks.

Your bigger problem is that when you do not board the outbound Madrid to Barcelona leg, the rest of your ticket will be voided when you are a no-show.

https://thepointsguy.com/news/hidden-city-ticket-crackdown/

You have already received the answer why this won't work, but I wanted to take a moment to commend you for asking this question before you put yourself in a difficult situation. I only wish more people would do as you did and ask in advance!

I hope you have a great trip.

Nonstop R-T's to Madrid are looking quite cheap these days. $400 to $550 from NYC in October, unless you are looking for Business Class seats. If you are looking for a relative bargain for these, try TAP with a connection in Lisbon.

1). If I were to buy the NY-Barcelona ticket with the Madrid stopover, could I deplane in Madrid with my carry on luggage?

Yes

On the return, could I board the Barcelona-NY flight in Madrid?

Yes--BUT, Only after you have purchased a new, one way ticket.