Your initial plan has only two full, non-jetlagged days in Barcelona. For me that would be too short. Barcelona has a lot of highly popular (yes, crowded) sights for which you need to purchase timed entry tickets in advance to avoid extremely long ticket lines. That makes it a challenging city to visit on the run, because you have to guess how much time you'll want to spend at each sight, and it's rational in that situation to pad the schedule every step of the way. These are the problem sights:
La Sagrada Familia
Casa Mila/La Pedrera
Palau de al Music Catalana (English tours can sell out)
The city has many other worthwhile sights, but they generally do not require pre-purchased tickets. How much do you want to see in Barcelona?
Zaragoza is a very interesting city. A lunch stop won't really give you time to see much; I think the single most important sight is Aljaferia, an 11th-century Moorish Palace. if not planning to allow hours of sightseeing time, I'd probably choose a smaller place where I'd be less likely to spend time in traffic or looking for a place to park the car. I don't have an alternative to suggest, however.
ViaMichelin estimates the driving time from Barcelona to Zaragoza at 3 hr. 14 min. The fastest trains take less than half as long--only 1 hr. 23 min. The train station in Zaragoza is a bus or taxi ride from the Aljaferia or the historic center (in the same direction as the palace, but farther). I don't know anything about the convenience of renting a car in Zaragoza vs. in Barcelona.
I found Vitoria-Gasteiz considerably more interesting than Pamplona, which seems to get lots of tourists because of Hemingway but really doesn't have all that much going on except during the actual running of the bulls.
The hilltown of Laguardia is very atmospheric and worth a stop if you are passing nearby. Check out the location of the parking lot(s) below the town. There's an elevator up to the historic area.
I preferred Bilbao to San Sebastian. The latter is a lot more touristy.
I'm not sure you'll get much benefit out of the car except for one or two days. There's frequent bus service between San Sebastian and Bilbao, and there are also buses to nearby towns from both of those cities. You don't have a much time, so your opportunities for rural exploration will necessarily be limited. Not counting the arrival day and departure day, your trip is only seven days long. The most I'd attempt in that time would be Barcelona, San Sebastian and Bilbao.
Have you checked flights from Bilbao back to the states? Can you get something affordable on a single ticket? I wouldn't want to start the day of my transatlantic flight in Bilbao if I was traveling on two separate tickets.