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.

I took my first trip to Spain in April. My flight arrived in Madrid and I took the train to Seville for 3 nights, onto Cordoba for 2 nights, Granada for 3 nights and Madrid 4 nights (also, with a day trip to Toledo). I'm happy that I spent two nights in Cordoba, rather than a day trip. The narrow alleyways and flower pot covered walls were beautiful, but I encountered so many tour groups during the day that it was nice to see them depart allowing a quieter experience in the early morning/later evening hours. The crowds were pretty heavy everywhere I went, but I would imagine in January there would be less tourists, and unsure about weather then.
However you choose to arrange it, enjoy! What a fantastic graduation celebration!

I'd steal 1-2 days from Madrid
Cordoba is pretty magical in the evening so it'd be nice to do 2 nights bookending one full day.
Alternatively, add a night to Sevilla.

I do agree with an overnight or two in Cordoba. I think Granada would be fine in two nights, especially to give you more time in Cordoba. I do like the idea of leaving Madrid to the trip end and going immediately to an Andalusia town. I never buy expensive train tickets in advance for my arrival day. I feel it’s risky. So go to Cordoba from the Madrid airport. Then Sevilla and Granada and back to Madrid.