Ok the POD for this is around 800 AD. In this timeline the Vikings hold and influence on the Irish expands to include the entire island instead of being replaced by the local tribes.
Would the Normans have an easier time ruling Norse then Catholics or would they be more difficult to conquer. How would the Norse religion stand up against Protestantism?
Would catholic Europe try to exert more influence if the Irish were not Catholic?
Given the fact that the Norse merged with the slavic population in Russia, and the French in Normandy, the same thing would probably happen in Ireland. Ireland is rather small, and has easily navigable rivers running throughout it, so the Vikings shouldn't have too much trouble conquering it.
If the Norse did conquer Ireland, and wished to keep order and control, converting to Christianity and respecting the monasteries would be essential. The Normans faced rebellions when they tried to use abandoned monasteries as quarries. Conversion would make intermarriage with allied irish royal families easier, too, and doing so would make claiming a title like "High King of Ireland" or "Emperor of the Gaels" more legitimate in the eyes of the local elites and the populace.
Gaels were very much land focused, with wealth being measured in terms of how many cattle you had. Since the Norse were mainly focused on the sea, i could see a united Kingdom of Ireland under a Hiberno-Norse aristocracy developing a sort of Hiberno-Norse cant for trade between Gaelic inlanders and Norse coastal towns. This would eventually merge with Irish as the inlanders migrate to the coast, with Irish being the predominant component, and Norse mostly surviving in urban slang and commercial terms.
I can't really see large scale Norse settlement happening inland, as it was already inhabited and rather less desirable than the south, east and south-east coastlines.
Hiberno-Norse trade with Britain and continental Europe would be larger, bringing in some foreign influence as well as foreign criticism of the irish interpretation of Catholicism. Either they "normalise" their church or become outcasts, and since the Norse don't want to piss off the monasteries more than they already have, they might keep the status quo for a few more generations before converting, without necessarily facing a crusade. In any case, a unified and relatively prosperous Ireland is less of an easy target than an extremely divided Ireland.
A united Ireland in the second millenium would probably play a very minor role if any in continental affairs, and would probably clash with the Kingdom of Scotland over the western isles and the isle of man. When/if the reformation comes around, they have a possibility of converting as Scotland did, perhaps by claiming to represent traditional Irish Christianity that was snuffed out by continental pressure.