Updated : 15/05/2013
NO - There are currently 27 countries in the EU, but only 17 use the euro: Austria, Belgium, Cyprus, Estonia, Finland, France, Germany, Greece, Ireland, Italy, Luxembourg, Malta, the Netherlands, Portugal, Slovakia, Slovenia and Spain.
YES - Duty-free shopping, for example in airports, is only for non EU residents whose destination is outside the EU. EU residents must pay VAT, even when they are shopping in another EU country.
You may sometimes be exempted from paying VAT when you buy a car in another EU country.