Is Mid September a good season/weather to travel in Spain?

September 11 is the National Day of Catalonia. Is a day when many families take to the streets with flags and festive resemblance, it pays to be in Barcelona on this date.

I think the itinerary is well designed and have time to do everything you have planned.

This looks good to me Note: You will never visit "all the nice Tapas shops in Barcelona" in one trip, or even two or three trips...but, make a start on them anyway!!
Sagrada Familia...get tickets in advance and avoid the long line..check the web site of the Sagrada Familia.
Sevilla...highly recommend Really Discover Seville for a good tour...and they can put you in touch with an excellent flamenco show...go early and sit near the stage.....alcazar..may be closed if the King is in town...check the schedule when you arrive.

Tourist traffic will be significantly down from peak periods of june, july and august. Temperatures will still be warm with less potential of hot. A good time to travel.
Trip recommendations:
- a day trip to granada is NOT a feasible expectation. It is a three hour drive each way. The Alhambra is worthy of at least a half day visit and tickets must be reserved and have specific time slots. Carefully research, via RS, Tripadvisor or other resources, how to visit alhambra. Ideally you would spend the night in granada.
- to max trip effectiveness you should purchase open jaw air tickets. Recommend you fly directly to Granada then bus/train to seville, train to madrid, train to barcelona and fly home from barcelona.
This means: overnight flight from usa arrive next morning in madrid then take flight to granada with arrival either early afternoon or late afternoon.
First night - spend night in granada and visit alhambra next day. Your challenge will be to get to hotel take a walk and visit part of old town. Then take nap, get up at 10 pm and go out to eat. To get the most out of spain you should live like a local. Stay up late and enjoy the night.
Second night - bus/train to seville (bus travel is comfortable in spain). Depart granada late afternoon. Arrive seville early evening. Nap on bus/train. Enjoy seville as you wish.
Third night in seville. After third night depart for madrid via AVE train. Purchase tickets on RENfe and save big bucks. Renfe is pain to use, learn how via web research.
On your way to madrid you will have the option to get off at cordoba for a few hours. Research and determine if this is something you want to do.
Arrive madrid and spend nights four, five and six. Choose between toledo or segovia for a day trip. Madrid is a loud city, choose wisely your room location.
After night six take AVE to barcelona for remainder of your trip. Spend half a day on a barcelona beach to bask in the sun. Enables some relaxation and provides opportunity to say you spent a day at the beach on the Med.

That looks like a nice itinerary and I'm glad to see you aren't rushing things or cramming too much in. The weather can be a little unpredictable in September but should be enjoyable (not too hot or cold). Happy Travels!