My thoughts.
I think you can see the big sights in 12 days. Remember you have travel days and tour days. Each full day of touring needs a night on either side, so two nights = 1 full day, three nights = two days, etc. Travel days are easily consumed with packing, checking out, getting transportation, finding meals, finding your lodging, checking in, unpacking and getting oriented.
I'm not a huge fan of Madrid (because it's a new city) but the Prado is world class and there are some really nice parks. I also liked Temple Debod (a little ancient Egypt in downtown Madrid). For me two nights is plenty and half my full day in Madrid will be spent in the Prado.
Toledo was my favorite twisty road medieval spot in Spain and your only taste of classical Castillian Spain (after this it's Moorish-influenced Andalucia). There are plenty of sights for another full day, so two nights. The train to Toledo from Madrid is from Atocha station (a short walk from the Prado). This is the train station where you can catch an AVE (fast train) to Cordoba after backtracking from Toledo.
For a short trip, a day stop in Cordoba (start early and give yourself four to six hours) will allow you to see the Mesquita and stroll the old quarter. You might consider starting in Toledo, then seeing Madrid and staying close to Atocha to save backtracking on a big travel day. Then get back on another AVE to Sevilla.
You will get to Sevilla late, so the day is shot, but you're only on night 5. Two nights in Seville will give you time to see the Cathedral, Alcazar and surrounding quarter during your one full day. I'd try to see a Flamenco show on one of your nights. There may still be bull fights in September if that interests you (may end in August in Seville, the season ends in October and the last corridas of the year are in Madrid). They start at 7 pm, as I recall, so plan dinner afterward.
From Seville, renting a car starts making sense but it depends on the options you choose.
You can train to Granada easily from Sevilla. Plan three nights in Granada, you will need a full day to see the Alhambra and another to see other great sights in town. Plan to see a Sambra performance (similar to flamenco) one evening in an Albaicin cave across the gorge from the Alhambra, stop to see the sun set on the Alhambra from the San Nicolas viewpoint in the Albaicin.
That's ten nights. If your 12 nights didn't include flights to and from Spain, this is really all the time you have. Fly home from Granada. Expect a connection, probably in Madrid.
If you still have two nights to play with. I'd probably go with renting a car in Sevilla and drive to Arcos for a look around a Spanish "White" town, then to Jerez by early afternoon and visit a couple of Bodegas (the only place in Spain we found Bodegas with regular public tours). Stay the night in Jerez then drive to Ronda the next day for a look around and one night there. Then to Granada, turn in your car on arrival. If you like old neolithic passage tombs, there are three good examples in Antequera. They're convenient by car as a day stop between Ronda and Granada.