Indeed, some will others won't, but the majority will because as I explained earlier, that's a day where a lot of Catalans from elsewhere come to the capital and they visit our museums as any other visitor would :))
If anything ensure you pre-book Picasso online because it's always very full, and that day more so. As per doing the tour... Barcelona has as many people in the street that day as it has many other months in summer, spring and autumn so I wouldn't worry too much. I know that Runner Beans for example they do offer their tours almost every day of the year, including Sep 11th, rain or shine, as their motto says. During that day people are spread across the city as there are different activities happening, and in the afternoon lots of them will gather for the rally in one of the main avenues of the city (Avinguda Meridiana) which stretches for over 5 miles and it's located to the east of the centre.
For shops that's another matter, except for a few souvenir shops I would expect the rest to be closed as if it was a typical Sunday. If anything, perhaps the Maremagnum Shopping Mall, located at the Port Vell (Old Port) area might be open as it's located inside the port and opening hours there are are governed by a different set of laws than those outside the port (that is, in the city). But again, I'm only guessing here.
As per restaurants, I hope to clarify some misunderstanding some visitors have: the city has over 15.000 eateries, some have their day off on Sunday, some on Monday and a few on Tuesday; some have one day off each week, others two. In short, you ALWAYS have a lot of restaurants open, maybe not the one you specifically wanted (if it happens to be their day off that day), but there'll be a wide choice to choose from.