Wow!! I have been to every place on your itinerary, and I think you have planned a perfect trip, especially since this is your first time to Spain. Congratulations on doing a lot of research! It certainly pays off.
I think you have the perfect number of nights in each place. For example, we had 2 nights each in Granada and Cordoba, and 4 nights in Sevilla, and I remember wishing we had an extra night in each of these places.
Regarding day trips, I think you will find that there is plenty to do at each destination so you don't necessarily have to do any daytrips, unless you really want to. Two good day trips from Barcelona are Montserrat (we didn't go there) and Girona. We spent 2 nights in Girona and loved it. But it is doable as a day trip from Barcelona. You could rent a car for the day; I think you can take the train from Barcelona; and there are group tours available, too. But you might not have time for day trips from Barcelona because you are arriving at night on September 1, so you might be jetlagged on September 2 and not able to do as much as you wish. Additionally, many sites in Barcelona now require advance purchase tickets. This means, depending on what is available each day and at what time, you might not be able to visit as many sites in a day as you would like. So you might need all 5 days in Barcelona.
I haven't been to Cadiz but I know it is a popular day trip from Seville, so that is a possibility. Keep in mind, however, that it will be the hottest in Seville, so you might not be able to do as much sightseeing in a day because of the heat. You should leave early every morning for sightseeing, and you may need to return in the afternoon to get some respite from the heat in your air-conditioned room. Better still, you may want to get a hotel with a pool. We were in Seville in late September/very early October, and it was typically 90F - 95F every day. So it might be hotter when you are there in mid September.
Segovia is a popular day trip from Madrid (I haven't been there). However, since you like museums, I think you might find enough to keep you busy in Madrid, which has some outstanding art museums.
BTW, I am happy to see that you are staying overnight in Cordoba and Toledo. Many people visit these 2 cities as day trips, and I think they are missing out on a lot. There is plenty to see in each city to keep you busy for a few days, and the evenings are wonderful after the daytrippers have left and the city is illuminated. Many people associate Cordoba with only the Mezquita, which of course is magnificent, but Cordoba has other wonderful sites to see.