Barcelona & Spain in early Oct.

That's a lot on your wish list for only 7 nights. The Pyrenees and Costa del Sol are opposite directions. The Pyrenees require a car, but you could see other places or even drive over to the Basque region (Bilboa and San Sebastian). Maybe enough time to hit a little of southern France.

Andalucía would be a 7 night trip all on its own. And Barcelona is not close so I really don't suggest that (or skip Barcelona).

Costa del Sol is mostly for sunning on the beaches. Not much to see and very heavily developed. Nothing wrong if that's what you want to do, but be aware.

Valencia is between Barcelona and Costa del Sol. Plus some other smaller cities. Plenty to see and do in those and you could make a decent 7 night trip out of that. And/or hit the beaches along that route rather than Costa del Sol.

Quite a hike to southern Spain from Barcelona..... Have you already purchased your airfare?
You could take the AVE to Madrid, spend two nights then head to Seville for two nights and work in a stop at Cordoba or Ronda. But most likely would need to fly back to Barcelona.
Open up google maps and start plotting the distances to better understand the task you have defined.

Another alternative, which makes more efficient use of your travel time, would be to take day trips from Barcelona to a number of local interesting destinations (Montserrat for example). Should you choose this option than recommend you rent an apt. October is a great time of year to visit.

We had a too-short time in Barcelona this past November and enjoyed our time there (including dinner at Tickets restaurant one night and 41 Degrees the next), but combined this visit with other locations in the northern half of Spain.

As noted above, Andalusia is a good distance from the North, and is a worthwhile destination on its own. Sevilla (especially the old town) is definitely worth visiting, and not next-door, but a lot closer than Barcelona, Valencia, etc. is Granada, with the wondrous Alhambra. If you haven't seen Sevilla and Granada (plus Ronda, Arcos de la Frontera, and more in Andalusia) consider a Southern Spain trip now or in the future. Including Barcelona with that trip would be possible with enough time and the expense of traveling between both south and north, but generally one region would keep a visitor busy for a whole trip.

October should certainly have comfortable temperatures in Sevilla and southern Spain, compared to the scorching summer.

If you really want to see Barcelona and a part of southern Spain in only 7 nights, fly between them (you can use Malaga, Seville, or Granada airport). Use http://www.skyscanner.com to look for flights.