Sevilla/Cordoba/Granada...how many nights?

Yes, more time in Seville.

You can find many, many posts regarding developing a itinerary for Andalusia.

Sevilla and Cordoba are amazing cities. I understand the nature of your trip with family in Madrid, but, you have a lot of time there, which as a newer Spanish city, is, IMHO, far less interesting than other cities in Spain. I always recommend overnighting in Cordoba. Its lovely in the evening and early morning without the day trippers. There also is a lot to see. Sevilla is gorgeous, larger city and has many attractions. We spent five nights and it was not enough.

I realize time is a very limiting but Segovia is a nice day trip from Madrid.

You do not need a car for any of this travel.

Have you found this on this website? Take a look at "at a Glance" for each city.
https://www.ricksteves.com/europe/spain/andalucia

Do one night in Granada, add a night in Seville and another in Madrid to day trip to Segovia.

I would lean most towards an extra night in Sevilla with a night in Cordoba as second choice. To me the issue is a logistical one. Sevilla is an easy add for a night, and imo two nights is not enough already. A one night stay in Cordoba just feels a little clumsy when it’s so easy to see en route from Madrid or as a day trip from Sevilla.

I took an early train from Madrid with a train to Sevilla late afternoon. I visited the Mezquita and Alcazar at a leisurely pace and still had time to explore a bit. That’s not to say I wouldn’t have appreciated more time, but I didn’t feel like I was missing out under those circumstances.

Grenada for me was kind of a tough one. I was there for three nights, but ultimately felt like that was one too many. Aside from the old quarter I felt it was a very workaday city, I found it less appealing compared to others.

Interesting, I hadn't thought about doing Cordoba straight from Madrid on the way to Sevilla. That way we don't have to do it en route from Sevilla to Granada.