Your comment about the lack of activity on this forum intrigued me. I am a RS acolyte. I've relied on the books, the videos, and this forum to plan our trips. I have been very pleased with the help I get here. But in planning our Spain/Portugal trip, I notice seemingly fewer postings than I'd seen for or about other countries. My impression, however inaccurate, is that there may be fewer "must sees/must dos" on the Iberian Peninsula compared to other parts of Europe, so fewer questions and fewer responses.
In terms of your trip with your twins... As I say a lot on this forum: we all have different tastes, values, patience, etc. Advising about your ten year-olds when I don't know them is brash on my part. We have been fortunate to travel with our children who still travel with us as young adults now. It certainly is a lot easier now to plan because they're, well, adults.
Your July 19 day, for example, in which you plan to walk to and from Sagrada Familia, Gaudi's other buildings, down Las Ramblas and then the Gothic Quarter is a good way to chase away the jet lag. That said, it is a LOT of walking for any age. And for ten year olds, I'm unsure that Sagrada Familia followed by the other Gaudi buildings will hold their interest. Las Ramblas and the Gothic Quarter may hold their interest. As adults, we walked about seven to eight miles every day, and the heat wasn't too bad for us in Barcelona. It can get uncomfortably hot, wearing the patience of you and your offspring despite everyone's best intentions and behavior. I suggest using MapMyRun or Google Maps to estimate how much you plan to walk and then ask yourself how much fun that would be for your crew in 90 degree heat.
My wife and I explored the Picasso Museum. It sounds as though your children are more interested in Picasso than I. One thing about museums: the "museum" walking pace as one wends through, reviewing the works, and listening to the guides can take its own physical toll. We were happy to stop for tapas and beer after our tour...
The cable car ride over the harbor is likely to appeal to your twins - but you have to get there. It's not exactly on the path to or from anywhere.
The beaches of Barcelona are pretty nice - not Caribbean beaches, but swimable if you're a pretty good swimmer. They're busy beaches - a lot of people and activity. And you have to be careful about watching your personal items (they broadcast a recording that advises against having anything of value on the beach). My kids at 10 would have been good with several hours there.
I'm not a Banksy fan. I personally find him to be a salesman of his own work and not the anarchist he portrays himself to be. That said if your kids like Picasso, a visit to see Banksy's work would make for good dinner discussion, comparing the artists and their intents...
Be aware that on Las Ramblas there are people - typically women - with balloons. They walk up to an unsuspecting youngster and hand the child a balloon who is thrilled to receive it. Then the woman who gave the balloon turns to an adult with the kid and demands money. We also saw on Placa de Catalunya people who literally handled youngsters as the people showed the kids how to feed the pigeons, which got the pigeons to sit on the kids' arms. These people also do it with adults, but kids are a good target for obvious reasons. These people encourage photos -- and when all is said and done, they expect tips. They are quite aggressive.
Last thought... I grew up in a military family. International travel is in the mortar of my foundation. We traveled with our kids for that very reason. Our adult daughter last night gushed about how appreciative she was that we traveled as a family. Whatever the challenges you face and experience traveling with your twins, the payoff for them as adults will be huge. My ultimate advice: Plan doing less with them and be more with them if you know what I mean and I think you do.
