Three weeks in Ireland sounds like Heaven. I hope you have dry weather. That's makes a huge difference.
I am assuming you have never been previously, so if I insult you, please forgive me.
You mentioned culture, history, etc. Don't forget music. Traditional Irish music can be found in many pubs and some of the performers are fantastic, playing multiple instruments, even if they are unknowns.
Driving a rental car is the only way to go. However, a warning - I'm sure you know they drive on the left. That takes getting accustomed to, especially if you have a manual transmission, but general competence comes fairly quickly. The thing to realize is that you will have no sense of positioning versus the centerline. THAT does NOT come easily. Concentrate on that. Especially on narrow roads, it is very easy to believe you are missing head-on collisions very narrowly when, if fact, you may not be. I have seen more tourists lose their left side mirrors to hedges or telephone poles than I care to remember. So focus on your position within your own lane.
From your sleeping plans, you seem to have a preference for large cities (Dublin, Belfast, Derry) unless you are just referencing the general geographic locations. Maybe you have a large family that can only be accommodated by hotels. But B&Bs are everywhere and I highly recommend them. Whether great or only fair, they are an experience in themselves.
You also seem to like islands. Maybe you are fishing enthusiasts. But island hopping can eat up time and they are generally small with limited attractions. I hope no one gets seasick easily. Remember, you won't get to them on a stabilized cruise ship. Personally, my one experience (one of the Aran Islands) was plenty.
When in Dublin, think about visiting Trinity College and seeing the Book of Kells. Nearby St. Stephen's Green is very nice in good weather. A side trip worth considering is to Trim Castle, largest of the Norman stone castles, used in the film Braveheart.
I'm not sure why you want to go to Cashel and then double back to Kilkenny. I would go to Kilkenny first (try to fit in the park at Glendalough along the way), then Waterford, then Cashel, before heading toward the Cork area. I have personally never liked the Cork City area. On the way to Cashel, maybe look in Carrick-on-Suir for a small crystal factory from workers who broke away from Waterford. Years ago we were able to enter the actual glass blowing area. (No OSHA in Ireland.)
On the SW coast, consider spending a night on Bantry Bay. Also, Bantry House is famous for its gardens. Between Bantry and Glengarriff, a drive inland (maybe 45 minutes?) will take you to Gougane Barra Nat'l Park. If the weather is good, it needs serious consideration for a 3 or 4 hour visit.
If you Drive the Ring of Kerry, drive counter-clockwise to avoid getting stacked up behind all the tour buses. The all go clockwise (or they did - maybe check on this.) If you sleep in Kenmare, I really liked the An Bruachan B&B. Coming into Killarney, Ladies View is a famous scenic spot. Killarney is (or was) the tourist mecca. Music and pubs galore. Whether you are religious or not, see the carved altar in the Franciscan Friary. Be prepared for dim lighting for photography. Muckross estate gardens are good and Torc waterfall is a 1/2 mile hike away.
Consider the route to Galway through Limerick (a lot of history) and Bunratty near Shannon airport. Home of the fully restored and furnished Bunratty Castle (Cheap daytime tour. The evening medieval banquet is pricey but the entertainment is very fine.), Folk Park and the world famous Durty Nelly's pub. B&Bs galore in the area.
I believe the Loop Head Cliffs are also known as the Cliffs of Moher. The Cliffs of Moher are a MUST see.
Running out of characters. More in another message.
A general comment: In my opinion the two most scenic counties are Wicklow and Donegal.
