7 days in Spain--need help with itinerary!

Feasible if you mean can it be done, but not if you intend to see anything but trains, planes, and hotel lobbies. Two nights in any local will get you just one full day of sight seeing. A travel day will get you at most a half day and an evening. I see one full day in in Madrid, one in Barcelonia and a bunch of days on trains and planes. Besides that relocating is tiring.

With just seven nights on the ground, I'd stick to one or two locations. You can buy yourself a day by by flying open jaw, i.e., in one city and out another.

I suggest,

Nights 1-4 Barcelona or Seville (really just two days touristing)
Nights 5-7 Madrid (really just a day and a half touristing)

If you went directly to Barcelona or Seville and stayed there nights 1-6 returning early morning to Madrid you'd get four full days in Barcelonia or Seville and a half from day in Madrid, a net gain of one day of real vacation time. You could spend that day doing a day trip from Barcelona or Seville.

Way too ambitious IMO! Pick 2 maybe 3 bases if open jaw travel. Remember every time you move you give up at least 1/2 day to travel & with only 7 days why waste precious vacation time.
Barcelona has a beachfront & boadwalk, great place to sit back, people watch & enjoy some white sangria. May satisfy your coastal need. Plenty of day trip options from Barc & Madrid.
Check seat61.com for travel tips. Renfe.com is the official train website for Spain & discounts available if purchased in advance. AVE fast trains Barc-Madrid in 2 1/2 hrs, Barc-Sev 5 1/2 hrs. Very comfortable. Definitely no to railpass!

Every day until #7 on your plan involves considerable travel. I would't consider it.

Some hard decisions need to be made, and warning, arguments may ensue.
A) Arrive Madrid. Direct to Cordoba (1N) > Sevilla (3N) > Madrid (3N)
B) Arrive Madrid. Direct to Sevilla (2N) > Granada (2N) > Madrid (3N)
C) Arrive Madrid (3N) > Barcelona (3N) > Madrid for airport (1N)

Your husband may enjoy the soccer stadium tours. We did Real Madrid's, spent 2 1/2 hrs, interesting even for non-soccer enthusiasts. We caught an Atletico Madrid game which fit our schedule. The fans enthusiasm was fascinating.

Too much. Since he wants Barcelona and you want south...do those 2.

Madrid is not as interesting, unless art is your passion. Its still great in a longer trip but you have to prioritize.

Transit - you simply don't have time for the off-the-beaten path things that a car would help with. Public transit will be much easier. No pass - just buy tickets.

Given your priorities I'd suggest
Arrive - train to BCN
2 BCN
3 BCN
4. Fly to Seville
5. Seville
6. Seville (day trip Córdoba by train)
7. Train To Madrid (early if you want a few hours to see the Prado, late if you want more time in Seville).
8. Home

Granada is my favorite, but you don't have time for it as well as Seville. You could do Granada, with a day trip to Nerja or Malaga instead of Seville if you prefer, but Seville is iconic and probably the best taste of the south.

Yes, your number of locations needs to be scaled back, so you're able to actually enjoy Spain. Here's my recommendation with the plane flight you've already reserved.

Day 1-Land in Madrid in a.m. and immediately take train to Barcelona
Day 2-Barcelona (day of getting over jetlag)
Day 3- Barcelona
Day 4-Fly to Seville
Day 5-Seville
Day 6-Train to Madrid in afternoon (3 hrs)
Day 7-Madrid
Day 8-Fly home from Madrid

I wouldn't bother renting a car on this trip - too much to see & do in these locations. Be sure to stay in the city center (check the RS book for specific area advice) to maximize your time in each location.

Interesting that I just posted my reply and see that Mira almost mirrors the same itinerary!

Two cities would be more enjoyable than three, and I prefer Madrid over Barcelona, but since you are set on Barcelona the 3/2/2 Barcelona/Seville/Madrid split is probably a good way to go (especially since the first day it a jetlagged throwaway). Barcelona is a pretty easy city to manage getting to and from the airport, the bus from the main square works great. The train between Seville and Madrid is a quick journey. If you cut back three stops it will still be rushed but not miserable. For a big city Madrid is very, very easy as well. Seville is a small city with the parts of touristic interest pretty close together. There is now a fast train from Barcelona to Seville, takes as little as 5.5 hours. You can probably get a flight cheaper, but definitely worth considering.

It's unfortunate that you're flying in and out of Madrid, when you really want to see Barcelona and Seville. But what's done is done. So, the itinerary laid out by Mira and Jean is the only one that will not have you cursing the entire time.

Do be sure to spend your last night in Madrid. You don't want to chance missing your plane home. If you want some time in a museum, arrive earlier in the day; if you don't, you can arrive later. Remember that dinner in Spain is 10 PM, so you can eat your last dinner in Madrid even if you arrive late.

As for wanting to see a coastal city, remember that Barcelona is right on the Mediterranean, and even has beach areas right in town.

The trains are much faster than driving on these routes, so definitely use the trains.

I would agree with others - I think you need to train to Barcelona immediately upon arrival and spend 3 nights there. You also need to pick a base for southern Spain, the one that appeals to you the most. We clicked more with Granada and found the Alhambra to be incredible. However, it takes more time to get there than Seville. Wherever you choose in southern Spain, fly there from Barcelona. If you choose Seville, it does have lots of wonderful things to see. We saw wonderful flamenco there. We hired Paul McGrath for a extremely well-priced tour of Italica; he also does day tours to the white villages. If you want to squeeze those in on your visit, could be an efficient way to do so without the hassle of having to rent a car or waste time getting around by public transportation. I'd spend your last night in Madrid.

Lastly, you will save money by buying train tickets from Sevilla to Madrid in advance. Until you get to the Renfe (train) office at Barajas airport, you won't know which train you can take to Barcelona, so just bite the bullet and pay another €30-60 per ticket and remember how much you saved on the flight. At least there are frequent trains so you won't have a long wait.

Mira and Jean offer an excellent solution, and yes shame your flights are thru Madrid, but ain't them deals GOOD !!