Hi Casey and Phil,
(Warning, long reply!). You have a lot of days in Barcelona, Madrid, and Seville. I love all three and they're good bases to explore the surrounding areas, but I'd build as much flexibility (in terms of bookings) into those times as you can in case you find out that one or other of the cities is not your cup of tea.
Starting in Girona is a great idea. It's a lovely, interesting, laid-back place to enjoy while getting over jet lag, and there are cool places like Besalù nearby. I wished I'd added an extra day. In fact, I know several people who liked the smaller cities and general region of Catalunya more than Barcelona itself.
Do you like wine? You might consider a stop in La Rioja or Navarre. I went through that area a few years ago and thought it was interesting and pretty, and I enjoyed the winery we visited (I'm not a wine connoisseur, beyond liking to drink it).
To me, two nights in Ronda and 1 in Arcos is too much, especially as you'll have about 10 other nights in Andalusia. (I should add my standard disclaimer that I appear to be the only person on the planet who found Ronda nice but overrated, ;)).
I think you can see/soak up the atmosphere of the "white towns" you're interested in two days and not have to rush about at all. Also, when I did the RS Best of Spain tour in 2016, we stayed in Arcos. It was clear that the town had suffered badly from economic decline, young people moving away, the effects of 2008, etc. Few businesses were open, I didn't see a lot of people either milling about or living their lives. It was really quiet, but not in a "slow down and savor life" kind of way so much as in a "halfway to a ghost town" way. This was in late June (maybe the weather was a factor), and the hotel was at the top of a big hill, but I can't imagine that Covid-related economic woes have helped matters since.
Have you already booked that flight from Barcelona to Granada? If not, you might trim the time in Barcelona. It's a fantastic city, but it doesn't sound like you are necessarily craving city-specific experiences. Maybe instead of picking up the car in Granada, pick it up in Barcelona and explore options between Barcelona and Granada?
I love Seville, but five days is certainly more than "necessary" to get a feel for the city, especially as you're also heading to nearby Cordoba for 3 days. On the other hand, at that point you'll be two weeks into your trip and ready for a brief "vacation from your vacation," and Seville's a very nice to just lounge around in as well. I don't think you'd regret being there five days, but if you find you "need" 1-2 days for someplace else, I'd cut a day from Seville and a day from Cordoba.
I'm not sure you need six days in Madrid, unless you're also using it as a public transport base and its accessibility to Toledo, Segovia, etc. Segovia is great, slower-paced, and with amazing views, so it might be worth actually staying there overnight.
Lastly, I have to put in a plug for the vastly under-appreciated north, though late September to late October is definitely going to be chillier and rainier. Bilbao is lovely, has the Guggenheim (for modern and not-so-modern art), and, if you've ditched the car by that point, has good air and rail connections.
Is this your first trip to Spain? Feel free to PM me. I'm not an expert, but I'm a major Spain enthusiast (I've been there five times) and always happy to go on (and on) about it.
Hope this helps!
Caroline