Rental car from Bordeaux to languedoc

I would keep the car until Perpignan or Port Bou; both have direct trains to Barcelona, but check schedules. Yes, Carcassone has only one train station; from there trains to Perpignan or Port Bou require a connection, except for the very early morning run. www.autoeurope.com is a good starting point for car finding.

Rick's France book covers routes either via Caunes Minervois and Minerve or via Payrepertuse and Queribus that would give you a fuller experience of Languedoc.

If you haven't been to Barcelona, I think it has much more of interest than anything between Carcassonne and the border. If you've been to Barcelona but not Girona, there's your answer. Cross-border trains on that line are infrequent, but there's lots of service between Girona and Barcelona, so you wouldn't need to keep one eye on your watch the entire time you were trying to enjoy Girona. The city has a charming old town (not small), a wall you can walk on top of, two historic churches and a great small museum: something for everyone.

The Deutsche Bahn has good schedule data. The fast AVE trains [edited to add: and the international train you'd transfer to in Narbonne] go to Barcelona Sants Station. The much slower regional trains make a few other stops in the city, one of which might be handy to whichever hotel you select.

If you're flying out of Barcelona the next morning (or, really, even if you're not), I think you'll find a hotel near Pl. Catalunya very handy. That puts you very near the Barri Gotic for an evening stroll, or you could go a few blocks in the other direction to see some of the modernista buildings along Passeig de Gracia. And you can get the Aerobus to the airport from Pl. Catalunya if for some reason your group doesn't want to take a couple of taxis.

When I use the deutsch bahn website, I can't find port bou or Collioure, France but can find trains from Perpignan, France to Barcelona. I thought port bou was in France, but now that I look closer I think it is in Spain (so I will go back into the website to see if I can find port bou or portbou, Spain to Barcelona). I wouldn't want to drop off the rental there as I believe there would be a huge drop off charge. I need to drop off our car in France as it costs way too much to drop it off in another country. Is there a town near the border but in Spain where I can drop off the car without there being a big drop off charge. I guess I am thinking of dropping it off in Perpignan, France as that seems like a place where I can do that and still pick up a train to Barcelona. Any other drop off recommendations in that area with good train service to Barcelona that would not incur a large drop off charge would be welcome advice.

We have not been to Barcelona but are spending 3 nights there and 5 in Madrid and surrounding small towns.

I'm not certain, but based on some Googling of "rail map France" I think Perpignan is your best bet. It appears that the fast train from Carcassonne/Narbonne goes through Perpignan and Girona to Barcelona. There's a second rail line that runs nearer the coast, from Perpignan to Cerbere (France) and Portbou (Spain) to Figueres and on through Girona to Barcelona. The trains along that second line are probably all considerably slower.

You're right that you probably need to drop your rental car on the French side of the border in order to avoid a costly additional fee.

Edited to add: I found Collioure on the DB website. It's on the line between Carcassonne/Narbonne and Portbou.

The DB site does reflect all of the towns you're considering. Be careful of spelling and use a date before Dec. 10. Collioure and Cerbere don't have direct service to Barcelona; both require a connection. You're right - Port Bou (the spelling used on DB) is on the Spanish side of the border, so that may change the price of the car rental. (Google Maps also shows a place with the same name near Perpignan, but it's not served by train.) Perpignan is looking like the most practical drop-off spot.