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1) Yes, Vueling transfer the luggage automatically for connecting flights according to question 7 in their customer service FAQ.
2) Yes, you should be able to get out of the airport and into the city. (You could do this even without a multi-entry Schengen visa, as you're not leaving Schengen in going from Italy to Spain.) All of those transport methods work, and they all take 30 - 40 minutes, so the best one depends on where you're going in the city. I would probably take the AΓ©robus as it goes directly from Terminal 1 (where Vueling operate from) to Plaza CataluΓ±a (at the top of Las Ramblas, within easy walking distance of most of the sights).