Andalucia for 13 nights and 12 days in September

I am not very familiar with Gaucin, so I will leave that for someone else. However, your idea of 5 Seville/4 Grananda/3 Cordoba is good. These are all longer times than is necessary for the main sights of each city, but will leave you plenty of time to linger. You will also easily be able to do day trips as desired. I might take Cordoba down by 1 day in order to spend a night in Ronda or somewhere like that, but that's splitting hairs.

Trains or busses between the 3 cities mentioned is good, but you might want to rent a car for a day or two if you want to take a side trip to explore some smaller towns. For example, leave Seville in a rental car and drive some of the white hill towns, and either return the car to Seville in a day or two, or even return it in Granada, and proceed to visit Granada. If all you want is to get from Seville to somewhere like Jerez, public transit is fine, but to go out into the countryside a car would be useful. Not for the full 12 days, but just for a couple days.

Other places to consider: Cadiz, Jerez, Gibralter (not worth it IMO but opinions vary), a day trip to Tangier, Morocco, or some time on the beach in Malaga or Nerja.

If you are starting in Lisbon, then flying to Seville makes sense. On the return, you could fly Seville to Lisbon like you said, or Seville/Granada to Madrid. It may (or may not) be cheaper to fly home from Madrid. Seville, Granada, and Malaga all have frequent flights to Madrid, or if you end your trip in Seville or Cordoba, Madrid is only 2-3 hours away via train. Do the math to see what is cheaper.

We spent a week in Gaucin in 2010 and really enjoyed it. It's a very sleepy little village that has great views and served us well for day trips to Ronda, Arcos, Gibraltar, etc. I'm not certain how you would get to Gaucin without a car. The streets in Gaucin are extremely narrow and some of them are one way. They made for some very interesting driving - especially when we went the wrong way down a one way street and ended up with a local man backing us out of the street :)

We rented our car in Granada after staying three nights (could have likely stayed one more and been happy) then drove to Gaucin. After the week in Gaucin, we drove to Sevilla - where we turned in the car. We stayed 5 nights in Sevilla and did a day trip to Cordoba.
I like your idea of staying over night in Cordoba but am not certain you will need three nights.
If you are going to fly from Lisbon, you could also fly to Granada and then make your way back to Sevilla before flying back to Lisbon.

Firstly make sure you head to Granada and not Grenada (the Caribbean nation Reagan invaded in 1983).
Travel by train will be fine if you limit yourself to just Sevilla, Granada and Cordoba.
However if you wish to add some out of the way 'hill town' locations then the case for a car becomes greater.
Curious to know how you came to wanting to include Gaucin.
Ronda is of course the most recognised of the white hill towns.
I might suggest a couple of days in Jerez which can be used as a base for excursions to Cadiz and Arcos de la Frontera.
Perhaps ....
Seville 4 nts
Cordoba 2 nts
Granada 3 nts
Ronda 1 nt
Jerez 2 nts

I recommend the 3 nights in Cordoba plan.. That gives you, at best, 2 1/2 days. You'll want to tour the Mezquita and walk around the old town, peeking into patios. There's a small local history museum, just across the bridge from the old center and a tiny ancient synagogue, both worth seeing. Another day you can go out to Medinat al Zahara, the summer palace of some of the caliphs. It's a sight the likes of which you won't see in many other western European places.

One thing - if you're not getting a car but want to go to some smaller places, be sure to check into busses. Some routes that are not possible on the train, or just at bad times for your schedule, may be better served by bus. ALSA is the bus company I've used in the region and I've always had good luck. http://www.alsa.es/en/