That's a lot of hotel moves for just eight nights. I wouldn't do it. Although I believe Chani when she says Cordoba is great for overnight stays, that's too many 2-night stops for me. I'd visit Cordoba on a day-trip from Seville.
Have you checked to see that there are still tickets available for the Alhambra on the day you plan to see it? I am doubtful. Which would be a shame, but if you don't need to go to Granada your itinerary will not be so rushed. A lot of us feel as if a day on which you change hotels doesn't really allow much sightseeing at all. So we'd look at your plans and say, "Four days in Spain for four cities. Good grief."
I have no experience driving in Europe, but I suggest that you use ViaMichelin or Google Maps to get at least an idea of the driving times for your proposed travel legs, then look at the train times on the Deutsche Bahn website. The worst divergence in favor of the train is the Madrid-Seville leg: Train takes 2-1/2 hours; car, 5-1/2 hours. And those driving times don't include any stops, getting lost, looking for parking. Or time spent at the car-rental counter, either, which reportedly is not the expeditious process we typically experience in the US. If you take the train, you can grab food ahead of time and munch along the way. That's not so easy to do--at least for the driver--when you travel by car. Also be aware that a car suitable for 4 adults may very well not have a trunk capacity adequate for their luggage. It's essential not to leave anything at all visible inside your car if you stop for sightseeing or food in transit between hotels.
Further thoughts: You may have to pay quite high daily fees to park in the cities you're planning to visit, and Easter celebrations may mean some streets you'd like to use are blocked off.
On the other hand, you are a group of 4, and if you are not ready to pounce the moment the train tickets go on sale, their cost may be rather high.
Anyway, here are the other estimated travel times (train vs. ViaMichelin's estimates, generally thought to be optimistic):
Seville-Cordoba is about 45 minutes by train vs. at least 2 hours by car (see above).
Cordoba-Granada is 2-1/2 to 4-1/4 hours by train vs. at least 2-1/2 hours by car (see above).
Granada-Madrid is 4-1/2 to 6-1/2 hours by train and at least 4-1/2 hours by car (see above).