These are some scattered comments before I head out for a busy last day of sightseeing in Bursa (Turkey):
You have a long list of sights in Barcelona, three potential out-of town trips (Girona, Montjuic and Colonia Guell) and daughters who enjoy shopping. Because so many Bacrcelona sights require timed tickets, its risky to try to cram a bunch of places into one day. Even though you have six full days, I think peeling off two days for Madrid will leave you short in Barcelona. I'm one of those who finds Madrid not as superficially attractive as Barcelona (it has no equivalent of Barcelona's Gothic Quarter or modernista architecture), though it's a very lively city with some fabulous art museums. (Barcelona's MNAC is good, too, by the way.) Then again, Madrid has the Palacio Real, which many tourists want to see. (I found it not particularly enjoyable, but I'm not into the sort of furnishings you find in most palaces.)
The other thing is that Madrid has a long list of exceptional day trips--destinations so good that they're worth a couple of nights each if you have time. There's also Zaragoza, which is between Barcelona and Madrid. You could make a separate holiday trip to Madrid and environs in the future.
I'd think long and hard before committing to including Madrid this year. Unfortunately, you'll sort of need to make that decision in advance in order to avoid paying for two hotel rooms on one or two nights. In addition, Barcelona-Madrid rail fares on at least some trains will go up as you approach your travel dates, and trains between Barcelona and Madrid can definitely sell out. I'd suggest watching Renfe.com to see what happens to ticket availability and fares over time. Look at dates in late September now and compare fares to dates well into the future.
Casa Mila, Parc Guell, La Sagrada Familia, Casa Batllo and the Picasso Museum all require timed tickets. You'll need to buy them in advance to avoid interminable ticket lines--and possibly being handed a ticket for 3 hours later or running into a sold-out day. Although they aren't likely to totally sell out weeks in advance, individual time slots (especially early in the morning) can sell out early. I recommend spreading out the sights I've listed over your available days so you don't have to guess how much of a gap to leave between the entry times. There are some high ticket prices in Barcelona; allow enough time to fully enjoy the places you go. Note also that you'll encounter crowds at those five sights, which means taking photos can add a good bit of extra time.
I'm sure the GoT tour could sell out, so if having the tour is important, don't wait too long on that one. There are fast trains to Girona that would be faster than a tour bus. Train fares could be cheaper for early purchase.
I highly recommend the Barri Gotic tour offered by the Tourist Office. They limit the capacity (so you'd need to book ahead of time), so the group will probably be a lot smaller than the "free" tour groups (for which you really should tip generously).
Colonia Guell is a cheap local-train ride away, then a walk, so choose a day with decent weather. The audioguide is useful. No need to book ahead, but obviously you should check the operating hours. It doesn't stay open particularly late even in the summer, and you'll need to turn in the audio guide on time. I'd suggest leaving this little excursion till later in your trip. I liked it a lot, but you have a large number of Gaudi sights on your itinerary, and this one will chew up a good bit of time.
You must make a point of seeing at least the exterior of the Palacio de la Musica Catalana, a modernista building not designed by Gaudi. Tours are worthwhile (the English ones can sell out), and there are some tourist-friendly performances there that give you a good look at part of the interior. This can be combined with the nearby Picasso Museum and/or the Barri Gotic tour.