Is it a good idea to travel to Barcelona during Día de la Hispanidad/Fiesta Nacional de España (October the 12th)? We plan to be there from 08.10 for a week, but are concerned that Fiesta can spoil our visit, as many places (mainly restaurants and attractions) might be closed. And I have heard that the Fiesta can spill onto more than 1 day, is it true?
Restaurants and tourist attractions generally stay open on public holidays, because there's extra trade.
As far as restaurants go, Barcelona has plenty of restaurants, so even if a few close you shouldn't have trouble. The thing you do need to be prepared for is that supermarkets will be shut, so if you run out of bottled water you'll have to pay through the nose at a bar or a small grocer.
For individual tourist attractions, you can check their websites. I expect most of them to have English translations, but if not then the words you need to look for are festivo (public holiday) and víspera de festivo (day before a public holiday - they sometimes have special timetables). Similarly for transport.
When people talk about the holiday "spilling onto more than one day", they're probably talking about the institution of puentes, whereby workers make a "bridge" between the holiday and the nearest weekend. This isn't really relevant to you as a tourist, except insofar as tourist attractions may be slightly busier than usual for weekdays outside peak season.