14 days in Spain

Why Andorra? It is mainly a discount shopping place, electronics, etc.
Spain's AVE, high speed trains, are excellent. Have you considered using them?
It will be very HOT in Southern Spain in August (Granada, Sevilla, Cordoba) Norhern Spain, San Sebastian, Santander, would be much more comfortable.
You could rent a car in Barcelona to travel to the Costa Brava which would be wonderful for your family to have some time by the water.
As far as luggage, if each of you bring a 22" bag, you should be able to fit them in a standard sized rental car. You choose the size car you want. Automatic transmission will cost a lot more. Look at AutoEurope website for information on amount of luggage each car can hold. Never leave any items exposed in car due to break ins. If getting a car, book with AutoEurope, a US based broker, too.

No, I don't think that is a reasonable itinerary. You have probably got less than one full day in Madrid, and you definitely have less than a day in Granada, Valencia and Andorra. I agree that Andorra is a curious choice if you mean the capital city of Andorra la Vella rather than just wanting to see some mountains.

If you're not arriving in Madrid until August 17 after quite possibly a sleepless night on the plane, I think it's asking a lot of a family of 4 to deal with the arrival airport, pick up a rental car, check into a hotel in Madrid, drive to Toledo, actually sightsee in Toledo, and drive back to Madrid. Few people would be safe to drive back to Madrid like that, late on their arrival day. (And this part of the trip could easily be done without a car.)

I believe you haven't taken a close look at driving times. You can use ViaMichelin to look at your proposed route, but allow extra time, because folks here say VM is too optimistic. You must investigate every leg of your planned journey, adding in time for checking out of your old hotel (if you're not just taking a day-trip), finding parking at your destination, and checking into your new hotel.

ViaMichelin says it will take at least 6 hours to drive from Valencia to Andorra and at least 4 hours to drive from Andorra to Tossa de Mar, a beach town I selected randomly since I don't know where you're headed on the Costa Brava.

I often visit places for half a day or even less, but I would have to have an incredibly good reason to do so if I was going to need many hours of transportation time or if the destination was a large city (time-consuming to navigate, and nearly always with multiple days' worth of sights that would have to be ignored).

One day is pretty minimal for Seville, and I think when you delve into your guide book you'll come up with far more things to do in Barcelona than can be accomplished in 2 days. I suggest that you start by dropping Andorra and Valencia unless the latter is being included because you think your children will like the science-oriented stuff. If that's the case, I believe either Madrid or Toledo needs to go. And even that may not open up enough time to enjoy the other destinations.

The schedule you propose ...
Madrid (2N)
Seville (2N)
Granada (1N)
Valencia (1N)
Costa Brava (5N)
Barcelona (3N)
... is just not feasible, to be bluntly honest.

Travelling from Valencia to the Costa Brava via Andorra, in one day, with stops for sightseeing, is an absolute physical impossibility.

Something like this is still rushed, but a step in the right direction.

August
16. Arrive Barcelona. Train to Madrid (3 nts)
17. Madrid
18. Day to Toledo
19. Train to Sevilla (3 nts)
20. Sevilla
21. Day to Cordoba for the Mezquita
22. Train to Granada (2 nts)
23. Alhambra
24. Fly to Barcelona. Pick up car for Costa Brava (3 nts)
25. Costa Brava
26. Costa Brava
27. Drive to Barcelona (3 nts). Return car
28. Barcelona
29. Barcelona
30. Depart Barcelona

It takes two nights to get one full day in a place. You need two nights in Granada with one day for the Alhambra. I would want at least 3 in Seville and Madrid, but you can't do everything. Basically a one night stop is only really workable in transit where staying one night and having a pleasant evening in a nice small town works. One after another means the trip is all about piling in the car, checking in and out of hotels and driving and searching for parking. I would break this into 3 and 2 night stops with only one or two one night stops in transit e.g. Ronda would work as a one afternoon destination. Cut Andora and probably decide what other stops can go to have enough time to enjoy the stops you have. Seville is nice, but you could cut that and add time elsewhere. Cordoba though is a must for at least an afternoon at the Mezquita -- it is jaw dropping and unique.

17 - land in Madrid (jetlagged?, sleep-deprived?), get through airport, transfer to Madrid hotel. See this city and and day trip to Toledo.
18 - pack, breakfast, pick up rental car, drive 4-plus hours to Cordoba. Then find parking, get to historic center to sightsee and lunch, then return to car, drive 2-plus hours to Sevilla.
20 - pack, breakfast, drive 3-plus hours to Granada, check in to hotel, lunch, sightsee
21 - pack, breakfast, drive 6 hours to Valencia

Note that the driving times (viamichelin.com) are generally optimal times in good conditions - no roadworks, no slow-moving vehicles, no heavy traffic, no wrong turns, and most importantly - no pit stops.

I wouldn't try this itinerary with a private guide and a private driver, never mind with 2 young kids strapped into a car for hours on end.

First of, thanks to everyone for taking the time and responding. The information is very useful, and I really appreciate it.

What do you guys think of an idea of staying somewhere in Andalusia (Malaga, perhaps) and doing day trips to Grenada, Seville and Cordoba from there? Then, drive to Costa Brava stay there with little excursions to Monserrat, Figueres, and then heading to Barcelona for the last three days of the trip?

Thanks.

Maybe you have underestimated the size of Spain with this itinerary, maybe if you had an extra week it would work with all them places, skip Andorra and Valencia you need more than a day for Granada and he heat will also have its affects as Andalucia is extremely hot in August

If tou want to base yourself in Malaga then look at flights to Barcelona with air vueling as with two young kids it is a very long drive and flights are cheaper

Go to ViaMichelin and check the driving times. Malaga might just work as a base for visiting Granada, but I think you'll find Seville and Cordoba are too far away. I do a lot of day-trips myself, but I don't like to do them to places generally considered overnight (or 2-night) stops. So I'd say OK for Figueres, Cadaques and Sitges, but not to Granada, Seville or Cordoba even if the geography were friedlier. Girona's on the borderline; I loved it and spent multiple nights there, but others have managed with a day-trip from Barcelona. However, because of its location, Giroba's a more convenient base than Barcelona for side-trips to Figueres and Cadaques, which argues for spending some nights there.

Malaga to Granada & Cordoba - about 2 hours.
Malaga to Sevilla - about 3 hours.
Double those times for return journeys.
90 mins each way is my upper limit for a day trip, but preference is for less than 60 mins.
Don't waste your time with so much driving. Split your time between Sevilla and Cordoba.

August
16. Arrive Barcelona (1 nt)
17. Fly to Granada (2 nts)
18. Alhambra
19. Pick up car. Drive to Ronda (1 nt)
20. Drive to Sevilla (3 nts). Return car
21. Sevilla
22. Day to Cordoba for the Mezquita
23. Fly to Barcelona. Pick up car for Costa Brava (4 nts)
24. Costa Brava
25. Costa Brava
26. Costa Brava
27. Drive to Barcelona (3 nts). Return car
28. Barcelona
29. Barcelona
30. Depart Barcelona