Latvia has been a member of the EU since 2004 (which entails all you mention), and now it joined the Eurozone as well, and will use euro as its currency.
EU countries other than UK and Denmark (who joined much earlier) are both required and allowed to adopt EUR as currency only as soon as they fulfill certain economic/fiscal criteria (don't have the exact figures in my head, easy to search though), whether intentionally or accidentally.
Actually no. The UK is special in that they do not have to join the eurozone unless they want to all other EU members have to join the eurozone once they fullfil the necessary criteria.
Edit: After looking it up, there are 28 countries in the EU and 18 of them are in the eurozone.