In those three periods of Irish culture, how did war and peace differ? How did the many Irish kings organize their kingdoms? Was religion different in Ireland than on Britain? Is the historical English stereotype of nomadic, lawless "wild Irishmen" unfounded? Lastly, the Statutes of Kilkenny prohibit the performance of Irish music. Does that refer to only music in the Gaelic language, or a genre that was musically distinct from English music?
Im working from home today so a lot of this is from memory. But here goes!
KINGDOMS
The concept of kingship in Ireland before 1200 was quite different from what would be considered the norm in Europe during this period. A king or Ri could rule a territory ranging from the size of half a modern Irish county to an entire province. However the structure was not formalised or legalistic in the fashion that was common in feudalism.
Rule in Ireland was organised along familial lines. The the most basic building block of social organisation was structured around those who shared a common paternal grandfather. Wider allegiance was also owed to those who could trace ancestry further back also though. The Ui Neill dynasty for example ruled much of modern day Ulster and Leinster. That is not to say that one single family ruled over this entire territory, rather it was a loose-ish federation of families with common ancestry. The high kingship would fluctuate between different families within this federation. There was little in Irish kingship that was static as ill explain in further detail in the next paragraph.
The Ri of East Breifne(Modern day East Cavan) for example might be completely independent, owe allegiance to an overlord of Breifne, swear fealty to a King of the Ulster province or even potentially recognise a high king of Ireland (In reality however no king ever actually exercised complete control of Ireland). The annals make little difference between in their recording between the small petty kings in Ireland and the larger provincial ones. This, as you can imagine makes researching Irish kings confusing. There is so bloody many of them.
So, in summation Irish kingship in the period in question was localised, family group based and highly changeable.
KINGSHIP
The Irish king of the Early medieval period had a fairly loose association with his subjects. Kingdoms were highly de-centralised. The king of an area would have had his own ringfort and surrounding lands were he drew his livelihood from. The current theory is that Irish kings practiced whats known as itinerant lordship. They would travel constantly around their vassal regions exercising their lordship and being feasted by their subjects. Taxation would be minimal and unorganised. Each settlement produced what they needed for their own consumption and gave some surplus to the king - he did not directly control production.
WAR
War during this period appears to be largely small scale and raid-like in nature. Cattle were hugely important to Irish society during this period. Legends like the Tain feature cattle constantly in the context of warfare. Levies would likely have been drawn by kings from their vassals and allies through their familial ties.
As to how battles physically went down or what warriors fought with we dont know a huge amount about. The spear and sword are important symbols in Irish mythology, and archaeological evidence supports the the likelihood that these were common weapons. Chariots also feature in the annals often, however it seems likely these were transport for nobles rather than an attack vehicle as in other cultures. Warfare was likely constant, disorganised, seasonal and built around raid and counter-raid. It is unlikely huge or formalised armies would have taken to the field.
THE VIKINGS
The arrival of the vikings on Irish soil had some pretty widespread effects on the landscape. However, it didnt fundamentally change face of Irish society. It is for this reason that the above description of Irish social organisation is relevant for most of the period in question. The vikings did have several important effects though.
Firstly they founded towns and developed a trade network in Ireland. The Irish, up to this point, didnt really have anything resembling towns other than ecclesiastical settlements. These didnt quite fulfill the role of towns, certainly in the sense of providing a trade hub. Viking towns became hugely important in Ireland, and remain so to this day. Nearly all large Irish cities are based on viking settlements. The increase in trade and wealth between the towns made them valuable assets which the local gaelic lords fought to control. Viking mercenaries also became valuable assets in the constant warfare between the Irish tribes.
Initially the Irish attempted to drive the vikings off the island and were successful in some cases. Over time the viking towns just became a part of the landscape, albeit a valuable one. There was, in reality, no concerted viking invasion of Ireland. The image of Brian Boru fighting the good fight against the terrible Norse is a part of Irish nationalist mythology. There were vikings on both sides of the battle of Clontarf.
Summing up on the vikings, they came, built some towns, fought some wars and then got pretty quickly subsumed into the culture and social structure of Ireland. The reason they are depicted as being a terrifying and dangerous plaque on the island is because Irish history was written by monks who were obviously not entirely enamoured towards the pagan vikings.
Religion
Irish Christianity did differ from that of the continent for several reasons. Firstly Ireland was never under the control of the Roman Empire. Christianity was a largely Urban and episcopal (Bishop) based religion during its initial years. This system therefore was not conducive towards its spread in a de-centralised society like in early medieval Ireland. This, combined with the retreat of papal authority from North-West Europe after the collapse of the empire, meant that Christianity in Ireland, Scotland and parts of Northern England developed in isolation. Some differences naturally developed.
The lack of town, as mentioned before, meant monasteries became the centres of Irish christianity. A different system of dating was developed also. The Penitentials, a series of handbooks written during this period, outlined the public and private punishments for nearly every sin imaginable.
Im afraid I dont know much about the Kilkenny statutes, they are not a subject I have studied. As to whether the Irish were wild and lawless im not sure, it depends on your definition. The stereotype of the wild Irish was propagated by a culture who wished to subdue Ireland so presenting its population as wild and unruly was convenient. Irish social structure definitely lacked some of the features of kingship and social organisation which developed in Anglo-Saxon Britain and under Norman rule but I dont know if id call it wild.
Primary sources include the Annals, the Legendary tales the translated Brehon law texts etc. Most important secondary sources: Ó CRÓINÍN, D., Early Medieval Ireland 400–1200. CHARLES-EDWARDS, T. M., Early Christian Ireland. I can provide more specific secondary reading if necessary.
I am a history student in UCD Dublin and the hypothetical Ri of East Breifne was one of my ancestors if anyone finds that interesting! Im at home today so I cant provide academic sourcing on everything. If you want to know anything else I can find it out easily enough. Hope you found my answer enlightening!