Madrid and Andalusia in April - Itinerary feedback/suggestions?

I'd swap the order for Granada and Sevilla, so you only have a short hop on Day 2.
I wouldn't even consider flying overnight to Spain, getting thru Madrid airport, then across Madrid, on a train to Cordoba to see the Mezquita, before finally making your way to Granada. I don't think you would physically have the time, let alone the energy.
Arrive in Cordoba and adjust from jet lag on Day 1. Visit the Mezquita the next morning before making the short hop to Sevilla.

April 16: Arrive in Madrid and head straight to Cordoba on High Speed Train - 1 night in Cordoba
April 17: Train from Cordoba to Sevilla - 3 nights in Seville
April 20: Train from Sevilla to Granada - 2 nights in Granada
April 21: Book well ahead for visit to the Alhambra
April 22: Train from Granada to Madrid - 3 nights in Madrid
April 24: Day trip to Toledo
April 25: Return trip home

The first thing to take into account is Feria, the biggest festival in Seville, probably in Spain, April 12-17, 2016. There will be bullfights during that week. Do you want to be in Seville then or not? If you do, you should go to Seville when you land. If you want to do this, make sure you have a hotel reservation.

If you have an afternoon or evening flight from Madrid, you can spend your last night in Cordoba and be able to start in Madrid. I do like the idea of going to Cordoba (or even Seville) when you land. You would take a local train from the airport to Atocha station (Madrid) and then the fast train. It's about 3/4 hour longer on the train if you continue to Seville.

I think your "night" allocations are good.

Thank you for the feedback. I was trying to avoid Seville during Feria. Sounds like this might be the way to go.

I think heading to Cordoba on arrival is fine. However, I would not try to see Cordoba and then get on to Granada on arrival day! Arrival day is often exhausting. Make your way from the airport to the train station, go to Cordoba, and have a low-key evening. See the Mesquita the next day, then do an afternoon bus to Granada. Make sure you get your Alhambra tickets well ahead of time for the 18 - they book quickly and that's really the only day you have. Sometimes these go on sale 3 months out.

Re: Cordoba - you will get varying opinions on this. Here is mine: The Mesquita is a world-class sight and you shouldn't go to Andalucia without seeing it. However, the rest of the town is interesting but not anywhere near as interesting as Seville or Granada, or other places. I would do it as a day trip from Seville, but the feria complicates things as far as timing (normally I'd suggest going from Madrid to Seville on arrival day, staying 4 nights, one day in Cordoba, back to Seville for dinner).

You could go straight to Granada on the 16, see the Alhambra on the 17, take an evening bus to Seville on the 18, stay 4 nights in Seville with Cordoba as a day trip, then on to Madrid as planned.

I agree with David's suggested itinerary. Head to Cordoba the first afternoon and spend the night. It's a great place for an overnight and will be a great way to start your trip. See the Mezquita the next morning and take a short AVE ride to Sevilla for your 3 nights. Sevilla is the place for flamenco and the Casa de la Memoria de Al'Andalus has the best in my opinion (the performers are more focused on the art form and not just a tourist show). Granada next - be sure to reserve your Alhmabra tickets as soon as they are available for your date. I like the morning slot so you can enjoy a leisurely lunch afterward and then spend time wandering the grounds or shopping before heading back into town. Finish with Madrid - the Prado and a trip to Toledo.

I would not try to go back to Madrid on the day of your flight even if it was in the afternoon. Too many things can happen with the trains to prevent you from catching your flight (the trains mostly run on schedule but accidents and weather events do happen). I would not spend your first night in Madrid and later come back. You may find it underwhelming the second time.

Tripadvisor is a good site for hotel and restaurant reviews and can be programed to be based on price. I almost always reserve directly with the hotel (some offer deeper discounts for non-refundable advanced payment).

Bus tickets can be bought as you go. AVE tickets (the highspeed rail) should be bought in advance to get pretty good discounts. Use Paypal to by the tickets off the RENFE website (don't even try using your credit card). The Toledo-Madrid tickets should be bought a day or two ahead (like when arriving back into Madrid from Granada) - not because of discounts but because of long lines at Atocha. That way you catch the train you want and aren't standing in line still waiting to buy a ticket as it departs.

Sounds like a great trip. Enjoy!

Douglas makes good points. Start in Cordoba, then Granada, then Seville (give them a couple days to clean up the city after Feria), then Madrid.