A lot of these ideas are ones I've had on my mind as well.
I definitely suggest absorbing the Armagh city state and it's Monastery improvement, and it gets additional faith/culture from being isolated from districts, and appeal (like a faith chemmamul), as opposed to the current "extra faith from adjacent districts" nonsense. That's exactly NOT what a monastery is.
The Unique Unit definitely leaves room for a few fun ideas:
First, the Kern, which could replace the Skirmisher, similar to the Waraq'aq. So, to differentiate it from that unit, we could add the ability for it to see through trees and be unimpeded by the movement of woods/rainforest, similar to the Ngao Mbeba (Kerns liked to strike from the trees in ambush/harassment tactics). They could also be slightly and weaker and cheap, since their preferred weapon was throwing darts. Although Kerns did fight with Gallowglasses, they were a very true Native-Irish troop type.
Also, there is the Gallowglass suggestion. However, they were more Western Scottish than Irish troops, and it would also likely give us another military tactics unit that we all are uneasy about due to hard-builds. It could be a "buy only" unit, but it still makes a bit more sense for Scotland. That being said, as Gallowglasses did later come to Ireland and become more Irish, it could be a very nice melee infantry, possibly have to be bought, but powerful. They could be kind of a dismounted Knight since many of them were landowners.
Lastly, the Hobelar was suggested, a possible Courser replacement. While yes they were used by everyone, so was the Hussar, and we've seen it appear as a Unique Unit twice (three times if you count the hidden Austrians in Civ III).
In short, all three are decent options, but I'd go for the Kern.
So we now have a unit, and an improvement.
Abilities?
Well, since we have the Monastery, let's go with a religious ability.
Celtic Church: +1 faith for the city from all tiles with high appeal. Receives twice as many great prophet points from Holy Sites and their buildings. Receive an extra belief when founding a religion (stolen from the Civ V Byzantines, inspired by the distinctive practices of the Celtic Church). Apostles have one extra spread and are 15% cheaper.
Now, a leader:
Let's go with Brian Boru
High King of Ireland: Cities with a Monastery receive +2 loyalty on your home continent. Gains +1 gold and faith for every city with your religion, increased to +2 if that city has a Monastery improvement. This gold is doubled if that city has a Holy Site in it. Naval Units and Melee units receive +3 Combat Strength when attacking a city, +5 on his own continent. Unlocks the Currach unit, which doesn't replace anything. It is available at sailing and is weaker than a galley...but is a coastal raider. This means it benefits from invisibility and the coastal raids, and upgrades to a privateer. It is a melee unit though.
So, I added a unit here, btw - returning all the way from Civ III if any of you remember...I added it for the nostalgia faction, and I also added it as a nod to Boru's successful incorporation of Naval tactics with his conquests. So, Boru has some light domination abilities, but really encourages religion with the rest of the Irish abilities.
Agenda: Cogadh Gaedhil re Gallaibh - Wants to control and conquer his entire continent. Dislikes those with cities on his continent. Likes others who aren't on his continent. Hates anyone who conquers his founded cities. (Referencing his attempts to unite and control the entirety of Ireland).
Capital: Caiseal