Is the European Community different from the European Union? From my part of North America, they both look the same.
The European Economic Community became the European Community in 1993 with the adoption of the Masstricht Treaty, which also created the European Union, of which the EC was one of the pillars. The European Community was phased out as a separate entity by the Treaty of Lisbon in 2009, which fully incorporated all of the functions of the European Community in to the European Union.
In short, what was the EC is now the EU.
Éireann is the genitive form of Éire, therefore it is used where you need the genitive form, in phrases like Bunreacht na hÉireann ("the Constitution of Ireland" or "Ireland's Constitution", but that Constitution is quite explicit - Éire is ainm don Stát. That means that you use just Éire when you are saying the name of the country, as in this exercise.
de réir Bunreacht na hÉireann "Éire is ainm don Stát" agus mar sin tugtar Éire ar an tír i gcomhaontuithe idirnáisiúnta.
Doesn't 'Eire' technically refer to the Republic of Ireland, as distinct from 'Eireann'- all of Ireland?
No. Like the English word 'Ireland', it refers both to the whole island and the 26 counties, just as 'America' refers both to the continent and the United States.
Ok but in English we have the phrase "Republic of Ireland" to differentiate the 26 counties from the whole of Ireland. Is there an equivalent in Irish?
However, it's worth noting that among natives Éirinn is the nominative now.
Is there any reason for that? All I've been able to find about "Éirinn" is that is' the dative singular of Éire. Not that that means much, as I'm still not quite sure what the dative case is.
Languages evolve. Gaeilge was once only the genitive form of the word for the language, but it’s now also the nominative form.
I'm not sure why they started using it; it's just what's happened.
And, yes, prescriptive guides (i.e. most of them) will say it's the dative of Éire, which it's not really used that way.
As for cases, I recommend Wikipedia.