IS THIS ITINERARY HUMANLY POSSIBLE??

At first glance it looks rough, but it's actually workable. The transportation time to non-sleeping stationary time ratio is high, but if that's what it takes to see what you want, have at it.

You could take a bit of the pressure off by skipping Tarifa. Tangier is atypical of Morocco, the ferry is the only reason you're going to Tarifa, ect. Spend a couple days in Gibraltar for a change of pace and ignore its detractors.

After doing a lot of research for my trip last year, I decided to skip Tarifa, Tangiers and Gibraltar for the reasons already given. There are lovely towns in Andalusia that you could add instead: Antequerra, Ronda, Jerez, to name some I liked very much.

I think a better train route would be Seville first, then Cordoba because the Seville-Madrid trains go through Cordoba. Also, why not go straight to Toledo from Cordoba, then spend your last nights in Madrid, instead of Madrid-Toledo-Madrid.

"Cultural highlights", recommend you include stepping away from tourist zones and invest time in being among the current culture of locals via walking among where they live, shop, work and eat/drink in non-tourist restaurants/bars.
In Madrid enjoy a picnic in Retiro Park and then take a stroll through the business residential zone north of the park.
You itinerary creates the opportunity to return with more memories about transportation then experiences with culture. You are also creating the potential for increased travel stress of getting from point to point and encountering "glitches" which could adversely impact the effectiveness of your journey.
Am I reading your comments to understand you will be flying to Barcelona via a connection in Madrid?
For fun lets count from Sept 18 to Oct 4 as being 17 full days to create a journey in Spain. and lets also pretend this is your first trip, but you will have the opportunity to make a second trip. So......
Arrive in Barcelona and stay five days as this is a big place with a huge menu of opportunities to partake. Barcelona has an excellent subway and you could also use the train system to take a day trip (I recommend Montserrat).
Take the AVE train (high speed, comfortable, ez to use and efficient with taking you to a station adjacent to the main tourist area of Madrid) to Madrid. Travel during late afternoon to enjoy the scenery. Online ticket purchase will save you $, purchase second class tickets and be surprised at how nice is this form of transport.
Stay four days in Madrid with a day trip to Toledo.
Ok, 9 days gone with 8 remaining.
Hop back on the AVE and scoot down to Seville. Hopping off at Cordoba is an option, but no real need to spend a night. Stay three days in Seville and take in a flamenco show (I advocate the Red Door).
Now sit back and recognize you have just experienced three vastly different aspects of Spain served in three very different "containers".
Five days remain and I want to be selfish by asking you to spend the last three of them in Granada.
So you have two extra days in which you face a number of decisions, but here is a tip.....use both days at the same location to reduce the "one night stands" (which I find to be weary and not very effective). For me Gibraltar is Meh! (It is a rock), but others can rightfully advocate why you should go. My choice would be to add a day to Seville and take a day trip to Ronda and stay four days in Granada and end your trip on a relaxing romantic note.
Travel back to Madrid on the evening of the 4th and stay at a hotel near the airport to ease your departure on the 5th.
A couple of tips.............
We prefer to stay in apartments, if we are staying a minimum of three nights in one place, and for a trip of this duration recommend you do so for the following reasons: better economics, more space to spread out, having a kitchen and you can find apartments with a clothes washer. Use Airbnb to get a sense of what is available in the destinations of your choice.
Pack way less then you are comfortable in bringing. Take the risk that what you leave behind will be found in where you go. Again, a trip of this duration will become a chore (especially if you stick with your original itinerary) of hauling your luggage from point A to B. Pack less, carry less, enjoy more.
Finally, get out a map and recognize how much geography you will cover via Barcelona, Madrid, Seville and Granada. The menu of experiences is vast, but............................. please take the time to enjoy the experiences which means.......... slow down!

Thank you to everybody who responded to this forum. All in all I am going to take everyone's advice and slow down and enjoy the experience rather than book as many places as possible. Thanks again.

Glad you've received some useful advice. Buen viaje!

For Gibralter make sure to check the current travel recommendation. I have been advised not to go there during my time here. Also, if you do go I have heard it is best to go with an organized tour group.

Also, when in Sevilla, or anywhere for that matter, you may get a better cultural experience if you go to more local tapas places. I stray away from touristy ones because I want to speak Spanish and have a true Spanish tapas experience!

WOW! This is a very ambitious itinerary! I would suggest following "Marbleskies" suggestions in order to spend more quality time in the cities you most want to visit and less time in transit in order to allow you time to appreciate the sites and the culture!
With the cost of transatlantic flights, (I live in Seattle) I can certainly appreciate the desire to include as much as possible but, I would advocate that the value is in the experience once in Europe; not the number of sites.
One of my most memorable experiences was sitting - just sitting - with a view of the Japanese bridge in Claude Monet's beautiful garden. My tour mates were in a hurry to decide where to have lunch. I told them to go on; I would catch-up. I can always grab a bite to eat; I may never again sit in Monet's garden. I roamed the garden, oblivious to the crowd, smelling the flowers and soaking in the experience.
I myself will be in Barcelona 19 SEP thru 23 SEP and have booked a "tapas tour" for 19 SEP and Spanish Civil War Tour for 20 SEP. I would be happy to share the contact information and to connect while we're both in Barcelona!

Marlene

Tangiers is kinda like Tijuana - not representative of the real Morocco. Save that for another trip. As for Gibraltar, while the rock is cool and all it is not worth eating up an entire day on a first trip to Spain especially given the current long waits at the border. Consider Jerez - the show at the Equestrian school is fantastic and you can visit a sherry bodega. Ronda is lovely too along with many of the hill towns.

Barcelona - 5 nights

Grenada - your current plan is two nights which really gives you 1 full day - you'll spend most of that at the Alhambra - you might add one night here

Tarifa - skip - add a hill town for two nights and the third to Grenada

Cordoba - 1 night

Seville - 3 nights

Toledo - 1 night

Madrid - 3 nights

Not sure why the Madrid - Toledo - Madrid unless you are working around a specific event.

We did a very similar trip but we drove it - Spain and its people are lovely - have a ball!

Thank you Marbleskies for the wonderful itinerary suggestions! This was very helpful to me as I am planning a similar adventure for late May/early June 2015.