As for the itinerary, if you're going to be in Cordoba around May 8-19 OR May 3 I would keep the 2 n

I'm planning a 15 or 16 night trip to Spain beginning the first week of May, arriving france and rental car in Madrid and returning home from Barcelona. We love to visit old towns/villages, eat outdoors, easy hiking. Coming from Miami, we're hoping for cool weather and no heat waves in Sevilla. Tentatively I have the following itinerary:
I know I am a broken record, but with that amount if time, you really france and rental car do not need a car. For example, you have two nights in Cordoba? That does not really leave much time for a day trip, and I assume you would spend the full day there at the mezquita and walking the town. There is so much to see in Sevilla, that I also don't see time for day trips, but if you wanted france and rental car to get out of the city you could rent a car for the day. There are good train connections between Sevilla and Cordoba as well as most Andalucian cities/towns.
Even without hiking excursions that would give you a more relaxed pace rather than going all the way to Valencia for one full day. You can fly to Barcelona, and I know there is also an overnight train from Granada to Barcelona.
Well if you want cool, maybe go to San Sebastian instead of going south - it took me many trips to Europe before I went in that direction and now I wish I had gone there much sooner. But it will not be the expereince of moorish spain that you find going south. You do not need to stay in Cordoba AND Seville - maybe base in Seville and then do a day trip to Cordoba. Barcelona again will be cooler than the south. Enjoy!
So perhaps it might make sense to take the train to Sevilla, stay there 4 nights and do a day trip to Cordoba. Then from Cordoba drive to Valencia. I really do like the freedom a car affords, and the rental cost is within our budget.
Since nothing france and rental car is written in stone yet, and we REALLY want to avoid the hot weather, any suggestions for cooler weather destinations in Spain? I've already been to Basque country and besides, we'd like to end up in Barcelona.
I've been to Andalucia several times in April and May and the weather has been quite comfortable - even sometimes chilly. It's quite similar to the beach weather in Southern California this time of year. I would certainly not let the weather bother you the first week of May.
I'm with everyone france and rental car else on the car - there's absolutely no need to rent a car based on the itinerary you've outlined - it will in most cases take longer to get from point to point by car and be a hinderance with trying to park and navigate in unknown cities. As BigRuss also points out - cost of toll, fule and parking can add up quickly.
As for the itinerary, if you're going to be in Cordoba around May 8-19 OR May 3 I would keep the 2 nights. france and rental car Most people only have time to visit Cordoba as a daytrip from Sevilla, which is doable. But Cordoba is a great city to spend a couple of days. Especially if your trip coincides with May 3 (May crosses festival) or May 8-19 (Patio Festival).
I would make a major change to this itinerary and skip Valencia. Instead visit Granada - it's a gem and I can't imigine skipping in lieu of Valencia. With Barcelona, Madrid, Sevilla, Granada and Cordoba it's still alot to do in 2 weeks but doable. Personally france and rental car I'd try to fly into Barcelona, then head to Madrid, Cordoba, Sevilla then Granada. Returning home with a connecting flight from Granada to Madrid then Miami.
You shouldn't be converned with the heat in the south in the first half of May. Past 16 years statistics for Sevilla (inland and usually among the absolute hottest in the country) the first two weeks of May says daily highs at 71-86F and night lows at 53-61.