Yes, more Itinerary help in Seville/South Spain September 22 - 26, 2018 (5 nights)

By all means, visit the Mesquite in Córdoba. It’s different from anything else you’ll see and it’s not a hard drive from Sevilla. Of course, the big draw in Andalusia would be a stay in Granada, with a visit to the Alhambra. Also not difficult by car. Personally I wasn’t thrilled by the hill towns. They’re charming, but it’s a long drive for not that much to see if you’re not headed to the coast anyway.

Rent the car as you leave Seville. Personally, I could spend weeks in Seville.

I would also recommend that you spend at least a day in Córdoba. It was the capital of the Umayyad Caliphate and is historically more significant than even Granada. Córdoba has one of the best attractions in Spain which is the Mezquita de Córdoba, it also has many other well-preserved Islamic architecture, as well as pre-Islamic Roman ruins. If you had more than 5 nights I would strongly recommend that you spend more than 1 night in Córdoba.

This is probably not ideal, but on such limited time I would do something like this:
Sevilla (3 nights)
train to
Córdoba (1 night)
bus to
Granada (1 night)

While in Córdoba do try the Salmorejo, which is the local dish of the town. It is a thicker, creamer, and (in my opinion) a more tasty version of a Gazpacho.

I'm sure Alhambra tickets are sold out, so if you decide to go to Granada, you need to explore the possibility of a back-door approach. Some people have had success with the Granada Card; others, by signing up for tours (be sure the Nasrid Palace is included).

The cathedral in Seville has the bones of Christopher Columbus and some other ancient artifacts. Across the square is the Alcazar. It's a must see in my opinion. My ex liked it better than the Alhambra in Granada. A walk down by the river to see the Gold tower (and the river) is nice. Consider it for dinner one evening. I think bull fights are done in Seville by then (but, I think, continue into October in Madrid if they interest you). I wouldn't recommend Plaza de Espana. It's an old fair sight, not particularly interesting or historical (nice gardens to stroll, not much more). There are Roman ruins outside Seville but we visited others elsewhere and didn't see these.

Other sights in the area. You can reach Cordoba by fast train for a day trip to see the Mesquite. It's pretty amazing and the quarter around it is worth strolling.

I like Jerez, which is a big bodega center in Spain. The only place I found that offered daily drop in tours. The drink is Sherry. Arcos de la Frontera would be a relatively convenient White Town to visit. We visited Ronda between Granada and Seville and liked it. I was disappointed with Cadiz. My FIL had said it was one of his favorite places, possibly because he went during a festival. It's arguably the oldest city in Western Europe, founded by the Phoenicians. It hasn't aged well, the sights don't really reflect the history and the city seems really run down. South of Cadiz are some great beaches. There are a couple of sizable towns just south, after those it's miles and miles of big deserted beaches.