It sounds like you've put two different questions into one. Your first question, in the title, is either "What was the political situation in Ireland like in the 11th c. C.E.?" or "What was the political relationship between England and the Irish like in the 11th c. C.E.?"

Your second question seems like it's "What are the roots of the conflict between Catholics and Protestants in Ireland?"

To answer your first question, Ireland was still largely ruled by a system of kings and high kings in the 11th century, although at that point Vikings had been terrorizing and occupying pockets of Irish land for several centuries. Strongbow was the first Englishman to conquer swathes of Ireland, about 100 years after the Norman invasion. The Anglo-Norman king, Henry II, was concerned about Strongbow declaring independence, but ended up allowing Strongbow to rule over "all Ireland" so long as he remained a vassal of Henry and not a king in his own right. In reality, Strongbow only truly ruled over a handful of Irish cities; however, Henry declaring that all Ireland was Strongbow's set up the later English conquest of Ireland and the long history of Ireland as England's first colony.

During the 11th century, there was no concept of "Protestantism" as we think of it--that began with Martin Luther about 500 years later. England became Protestant during the rule of Henry VIII, and although it was controversial, Elizabeth's reinforcement of Protestantism helped it to become the norm in England. Scotland also became Protestant, although a little later, and not the same variety of Protestant as England. However, Ireland remained primarily Catholic. England's rule over Ireland, and the import of Scottish Presbyterian settlers into Northern Ireland, meant that Protestantism was heavily favored legally, and Catholics, being native Irish, were oppressed (both for their birth and their religion). Catholic children had to go either to Protestant schools (at the risk of their souls) or simply not attend school. To be honest, England committed absolutely horrible atrocities against the Irish, and I personally take the Catholic, Irish, pro-independence side.

There is so much history and trauma there, but to make a long story short, native/ethnic Irish, who were mostly Catholic, felt oppressed by the English, who were mostly Protestant, and so the debate over Ireland's independence fell mostly along those lines--Protestants (some Irish, but many descended from English and Scottish settlers in Ulster) wanted Ireland to remain with the U.K., while Catholics wanted independence, both in order to freely practice their religion as well as simply to have self-governance.

Here is a secondary source that gives more details!

It should be added here that the ‘Norman’ period (aka 1066 until 1171 when Henry II turned up) basically saw zero English political influence upon Ireland.

Indeed one could argue that in those 100 or so years the Norman State was much weaker geopolitically than the ones found across the sea in Ireland.

Where does one get that idea from? Firstly it was William’s inability (or unwillingness) to maintain the Saxon scipfyrd system of ship levy. Or put much more brutally- the Saxon state maintained a large and powerful fleet which kept the balance of power in the Irish Sea previous to 1066. Afterwards?

The Irish saw the vacuum left by the English and filled the void.

Yet here we have to also pause- because the very terms ‘English’ and ‘Irish’ are not set the way later generations would like us to believe. When we talk about the English do we mean the Saxon’s, especially those dispossessed by the French speaking Viking descendant Norman’s? By Irish do we mean the Norse stronghold of Dublin or the native pre-Viking Powers?

Certainly by the 1070’s Dublin begins growing its fleets, expanding their influence greatly. Perhaps even earlier. Harold Godwinsun’s sons escaped the defeat of 1066, relocating to Dublin and it was from here in 1068 they raided the coast of Somerset before returning a year later to do the same around North Devon.

By 1075 the Viking’s of Dublin use of large fleets for force projection caused a naval arms race, and native Irish kings began constructing large fleets based upon Viking design in response. The only powers who could compete at the time were the Scottish Isles and the Isle of Man.

The immediate political impact of this can be seen if you follow the adventures of the half-Welsh, half Norse-Irish Prince Gruffydd ap Cynan. From 1075 to 1099 he raised no fewer than five fleets in Dublin (where he was born ironically) in order to claim control over Gwynedd. And his actions led other Welsh princes to do the same.

But while the dynasty of William of Normandy never displayed any grasp of understanding the importance of naval power, some of his fellow Norman’s did. The conquest of Pembroke was a harsh education for many of the Norman’s who attacked Wales. In 1094 for example, Welsh controlled Irish built ships allowed the Welsh almost drive the Norman’s out of the country.

Even more brutal for the Norman’s were the events of 1098; the Earl of Chester, Hugh the Fat of Avranches, was prosecuting his campaign against Gruffydd ap Cynan in north wales. As always Cynan was able to remain mobile and supplied due to the use of a Dublin originating fleet. Cynan had retreated up to Anglesey when his allies abandoned him, which forced him to flee back to the safety of Dublin to regroup and strike again.

If it wasn’t bad enough that the Norman’s had no way of countering this strategic defence in depth, just as they went to consolidate their power in Anglesey, a cruising fleet of King Magnus Barelegs of Norway was sailing nearby (they were in the region as the Norwegian King was asserting his authority of the Scottish isles and Isle of Man) and entered the Menai Straits. The resulting fight went badly for the Normans, the Earl of Shrewsbury killed by a Norwegian arrow and they were forced to retreat.

Via their allies in Ireland, the Welsh were able to retain control over their lands for at least another generation. In time this radical status quo was embraced by the Norman leaders of Pembroke; they were effectively cut off from the rest of the Kingdom by land, and existed as a semi-autonomous ‘island’ that needed its own fleet to stay in contact with the rest of Norman England.

As you can see, Ireland was the dominant power between the two nations; the fact that Ireland was utterly divided and engaged in much internal warfare during the period is perhaps the only reason why they were unable to exploit Norman weakness at this time.

Further proof of this is found by the alliance of Irish King Muirchertach O’Brian (who controlled the great fleets of the Viking ports of Dublin and Waterford) and Arnulf de Montgomery of Pembroke. O’Brain’s aid was crucial for Montgomery to lead Pembroke in rebellion against Henry I in 1102 and even though Henry was able to defeat Arnulf in a ground campaign, he was unable to prevent him from escaping to Ireland.

Increasingly Ireland became a sanctuary for those Norman’s who wished to escape royal control (and it must be said failures of royal control, such as during the reign of Stephen).

Many of these new arrivals nominally identified as ‘English’ (with the caveat most spoke French) and began to buy up land and holdings in Ireland (leading to the first complaints by Irish of English buying the best lands or subverting the native culture, a culture which had held off the Vikings to a large degree).

These were not military invasions at all, more a slow migration from the increasingly populated England and was part of a westward movement of such people into South Wales and lowland Scotland during the same era.

By the time of Henry II the situation was not improved at all. And the ‘invasion’ itself almost never came about.

Firstly, the main instigator of the whole thing, Richard ‘Strongbow’ FitzGilbert, had been Ruler of Pembroke, and as such was well versed in the geopolitical realities of Irish Sea power. Frustrated at his lack of success in Wales, he had relocated to Ireland and had become the son-in-law and successor of Diarmait MacMurchada of Leinster. He was t alone in doing things like this.

Another English adventurer, John de Courcy, had married the daughter of King Godfrey of Man and had began to carve out his own Man supported territory in Ulster (which allowed the Manx fleet claim control of the strategically important Carlingford Lough).

Yet it was FitzGilbert’s clear attempts to create a powerbase outside of Henry II’s direct control that the great King found so alarming. But even then Henry was in no position to stop him.

Previous to 1171 Henry had showed his own failure of grasp the importance of Irish sea power; his failed campaigns into Wales in 1157 and 1165 showed that he only saw things in terms of land battles. The Welsh willingness to use the sea constantly outmanoeuvred him and almost led to his death. Still, the 1165 campaign saw him hiring a Viking squadron from Dublin which was a start.

In truth Henry was only able to invade Ireland in 1171 because of the Irish. It was FitzGilbert’s rival, Rory O’Connor of Connaught, who needed Henry’s help to deal with Strongbow. It was this invitation, coupled with O’Connor’s alliance with the rulers of Man and the Western Isles (who supplied the all important ships) that allowed him cross the Irish Sea and claim the title of overlord.

This began the political force behind the demographic reality; Ireland had missed the window of opportunity to become the dominant power in the west; and while it is often seen as the start of the English invasion of Ireland, in many ways it should be argued as the end of Irish control over the west.

Hope that helps.

Source: Rodgers, N. A. M.; The Safeguard of the Sea: A naval history of Britain 660-1649; 1997; HarperCollins