Heading to BCN next spring. For flights from the US, we've found that Delta/KLM through AMS worked best for us, but we're in the western US. I believe there's a few direct flights from JFK and other east coast cities. Your plan sounds good, not too much. If you want to see the Sagrada Familia church, you need online tickets/reservations. If you love Gaudi, see some of his other works afterwards, and let your companion sit in one of the numerous cafes while you tour.
I've heard food tours can be quite fun as well, and might appeal to both of you?
We'll only have 5 days, but are planning to visit Montserrat and the surrounding wine/cava country for one of the days. Bus tours are an option for that as well, if you don't want to take the rental car there. Many Catalonians will not see themselves as Spaniards, so it will be quite a different experience from Madrid. Enjoy!