Jennifer, have you already booked for Spain? I was in Spain last May and went to all the places on your list except for Malaga.
The Oasis hostels in Granada and Seville are quite nice, but the doubles in Granada are inside a dorm with 6 beds. You get a private room, and it's locked, but you'll still have to share bathrooms with other people. The Oasis hostel in Seville went through some renovations last year when I was there but I imagine that they would be done by now. Try to request a room in the new building instead of the old building. I think the doubles in the new building are private, but the doubles in the old building are again, in a suite with a dorm room. The oasis hostels are a great deal- try to go on the tapas tour!
In Barcelona, I stayed at www.hostallevante.com last May and had a double with ensuite bathroom and balcony. It was really nice and the room was a big corner room. But I also had the displeasure of staying in a double without bathroom. The room was right behind the reception and the door was really paper thin- you could even hear the phone ring at reception! So overall, it's only worth it if you get a corner room.
In Cordoba I stayed at Hotel Los Omeyas. It's right next to the Mezquita- a bit pricier than your typical budget accomodation, but a nice clean place and good location.
In Madrid I stayed at www.hostal-palacios.com. Bathroom was small but my room had a tv! The hostal was close to the subway, which is a plus.