I'd have thought "I like the Irish entry" should be Is maith liom an iontráil Éireannach! I expect it's because the na makes it "the" entry, rather than "an" entry. Kind of like Poblacht na hÉireann is "The Republic of Ireland". It does accept "I like the entry of Ireland" though, which is more literal.
Yes, see my comment above.
In summary: Iontráil na hÉireann = "the entry of Ireland", in the same way as Poblacht na hÉireann = "the Republic of Ireland". The "the" is implied - "the X of Y" = "X na Y".
Remember that Irish expresses possession using a genitive case, rather than a possessive case. So "X na Y" is the equivalent of both "X of Y" (a genitive phrase in English) and "Y's X" (a possessive phrase in English).