Very doable, but I suggest you go to Madrid before Málaga so you don't need to backtrack to Madrid. High speed AVE train from Valencia to Madrid and from Madrid to Málaga. Huge international airport in Málaga.

Córdoba is perfectly doable as a daytrip from Málaga, 50 mins on the high speed AVE. Sevilla would be 2 hours each way, and I suggest you stay at least one night here.

You could perhaps do it this way:

Barcelona 4 days.
Valencia 2 days.
Madrid 2 days.
Madrid to Córdoba (daytrip) in 1 h 40 mins.
From Córdoba to Sevilla in the evening in 50 mins.
Sevilla 2 days.
Sevilla to Málaga in 2 hours.
Málaga 3 days. (and if you have to go back to Madrid, it's only 2h 30 mins by the AVE train).

You will not regret going to wonderful Málaga by the sea in June, but not only for the sea breeze and cooler temperatures. Genuine Andalucian atmosphere, great tapas bars, restaurants, cafés, nightlife, cultural scene and full of 3000 years of history and culture. One of my absolute favourites cities in the country.
http://www.telegraph.co.uk/journalists/nigel-tisdall/8773310/Malaga-Spain-Secret-Spanish-pleasures.html
http://www.guardian.co.uk/travel/2009/jun/21/malaga-spain-city-breaks-travel?INTCMP=SRCH

Go to Pedregalejo just east of the city center for nice beaches and lots of great seafront restaurants that also are very popular with locals.

The spectacular Nerja Caves are a little more than an hour bus ride to the east of Málaga city: http://www.cuevadenerja.es/index.php?lenguaje=en

Very popular La Marina in Nerja is as good as you get a seafood freduría in all of Andalucía: http://www.guideofnerja.com/pages/gutierrez/marina_en.htm
http://www.tripadvisor.co.uk/Restaurant_Review-g315917-d1002965-Reviews-La_Marina-Nerja_Costa_del_Sol_Andalucia.html#REVIEWS

I heartily agree with Kimhe about Malaga. Wonderful!

I think you're trying to squeeze in to much with the time you have for your trip. 2 days for Madrid will be just 1.5 days after travel time - barely enough to see parts of Madrid and no time for Daytrips. 2 nights for Sevilla is woefully short (and i wouldn't want to do as a daytrip from Malaga) and a daytrip to Granada will just be enough for the Alhambra and perhaps a quick visit in the Albacin. Frankly I'd drop Valencia and add the time to Madrid or Andalucia. If I had 7 days in Andalucia I'd spend 4 in Sevilla, even though it will be hot during the day - sightsee in the morning, rest/relax in midday and enjoy the evenings.

Doable yes but I wouldn't want to do it. I would skip Valencia, spend 3-4 nights in Sevilla, 1-2 in Malaga and 2 in Granada. End with 3-4 in Madrid.

FYI the schools break up in June too, so that may affect when/ where you go and availability.