Help planning Andalusia trip

I used an apartment in that area in November. I would certainly recommend the area for your time in Madrid. About 20 minutes walk to train station, several restaurants and depending on your jetlag, you can easily walk to the Reina Sofia museum or the Thyssen. I loved the Thyssen. For a private collection, it blows the mind. 3 floors of art collected by one man. His wife continues the purchasing with more modern pieces in annexes on each floor. There may be art overload however, when also considering Prado. I spent 7 hours there and pretty much saw every piece of art. The Reina Sofia offers Guernica. Some will say everyone should see it. It didn't move me as much as the other surrounding art and photos of the same time period did. Sobering.
When your eyes can no longer focus on fine detail, there is a pleasant Botanical Garden with green houses right beside the Prado. Retiro Park is a ways further.
So, you are flying out of Seville on 2/27 or somewhere else?
You can easily do a day trip to Cordoba from Seville and I recommend 2-3 night in Granada. If you can time it to be there on a Friday or Saturday night for a night visit to Alhambra, I would recommend it.
If you needed to be somewhere else to fly home, then there is a Seville-Granada-Cordoba-? option.

Tickets to the Alhambra often sell out well in advance, so keep an eye on tivket availability as you firm up your plans.

Both times I was in Madrid I stayed at Hotel Europa, which RS also uses for its tours. From there, it's a 15 minute walk downhill to the Prado, so slightly longer returning, or a short taxi ride. The Europa is on the Plaza del Sol, with easy metro and cercanias (suburban metro) access to both the train station and the airport. It's in the center of town. With luggage, use the cercanias - there's an elevator to street level. Taking the metro, you'll have to walk up one flight of stairs. The area around the Prado is fine, but I think Sol has more options for eating plus the convenience of the metro. The HOHO buses start at Sol so you may want to ride around and get an overview of the city on your first day. Check out the San Miguel market for grazing on tapas.

You may want to visit either the Thyssen (mainly Impressionists) or the Reina Sofia (20th c.) on your first day. They are much smaller than the Prado.

If you are flying round-trip Madrid, consider taking the train to either Sevilla or Cordoba on the 18th, then spending your last 2-3 nights in Madrid - that's if you are planning to return to Madrid on the 27th for a morning flight on the 28th. Saves one hotel change. Assuming you're flying out on the 28th a.m. you have 10 nights.

Cordoba (2N), morning train to Sevilla (3N), train/bus to Granada (2N), train/bus to Madrid (2N). Cordoba is my favorite Spanish city. It's quieter and smaller than Sevilla, a good place to relax and wander the well-preserved medieval Juderia for 1.5 days, see the beautiful, unique Mezquita and sample the city's small charming museums. Catch an early train (it's less than an hour's ride) to Sevilla and you'll have the better part of that day plus 2 more to enjoy, perhaps a day trip to Jerez and/or Cadiz or Arcos. Then train/bus to Granada in the morning, spend the rest of the day seeing the sights, your full day there at the Alhambra. Morning train/bus to Madrid, and 1/2 day for one of the art museums, and full day at the Prado. That gives you one extra day/night. I'd probably add it to Sevilla for more of that city's sights.

Hotel Mediodia is across from the main railroad station in Madrid. Convenient - if you have to drag suitcases - and it is almost "elegant". In short walking distance to the Prado.
However, my [alas late] recommendation is to visit Andalusia - particularly Granada - during Easter.