I would say that the verb here in English could also be in the plural "Ireland win..." as in this headline from RTÉ: "Ireland win Six Nations title as England fall short on dramatic final day", where both "win" and "fall" are plural although their subjects are grammatically singular.
It is never "EIRE", it is only Éire with an accent on the fIrst É. That accent isn't optional, as eire without the accent is a completely different word.
Éire is the official name of the country in Irish. "Ireland" is the official name of the country in English. EU documents written in English only refer to Ireland as "Ireland" - see the list of member states on the English language version of the EU website, versus the same list in Irish.
Irish is legally the first official language of Ireland, and the Irish language name is used on the passport, on stamps and on coins. While certain other Irish words are used in English (Taoiseach, Dáil, etc), Éire is not used in English.
You have to add the Irish keyboard and switch to it for this to work (at least in Windows). For US users, using the default ENG-US keyboard, Alt-Gr is the same as the other Alt key - it won't give you a fada! But it's definitely worth adding the Irish keyboard - it makes it much easier to complete the exercises!
Really? It doesn't get any more informal than Twitter and I see gaeilgeoirí using Éire all the time. I doubt it's simply because it's fewer characters.
Keep in mind also that it's hard to say anything like that is the rule of thumb for all speakers, giving the divergences in colloquialisms among the dialects, between older and younger speakers, and even among Gaelteacht speakers and urban speakers.