First pass at 28 day itinerary

Our interests are not too different (my wife is into birding), but our preferred travel style may be. Our first trip was all one night stays. Our second trip was one night stays except for 7 nights on the Shannon (LeBoat), and our third trip was a few two nighters, three in Dublin, and the rest one nighters. We travel very light and would much rather checkin/out of lodging than backtrack. We'll be in Ireland going the other direction just before you in 2023. Although you aren't detail specific, I'll throw in a couple of favorite places. We also ALWAYS stay in the lowest cost non-hostel lodging (I'm of an age that ensuite is essential and snoring keeps me awake) and it must be in walking distance of pubs claiming live music. Live music is iffy in May/June and many places are only once a week until high season.
Here's a couple of thoughts:
1) Flying in from SF, one of your Dublin days will be a fog of tired. I think there's lots of things to do in Dublin, especially great music. Two two-night stays may end up as one good day at the beginning on only one good day at the end, with travel etc.
1a) Include Brazen Head Sunday session (3:30pm) if possible. The Cobblestone has music every night and most afternoons. We've always avoided Temple Bar neighborhood, although it's probably good music.
2) I'd pick up a car as you leave Dublin and keep it until you return to Dublin. Learning to drive as you leave Dublin should be good enough, but that's your call. You could take a taxi to DUB and miss the worst of the maneuvers. Turning in a car, meeting airline checkin requirements, flying, claiming your bags, then renting another car will take longer than just driving. It's a small country. The drive from Dingle to Donegal is your longest, and there's a few things to break it up, like Galway, Doolin, Bunratty, InishMor.
3) Include Drombeg Neolithic site
4) Cahersiveen is right next to Skellig Michael. The puffins are in full attendance in May/June.
5) Consider InishMor overnight from Rossaveal
6) Skip Cliffs of Moher since Cliffs of Kerry are quite similar and uncrowded and you'll be driving by them on the ROK.
7) Consider Carrowkeel Tombs in Roscommon. You'll be crawling into mini-NewGranges all by yourself.
8) Bunratty Castle and Folk Village is sort of a hoot, although it's on the tour bus route (like Moher, Waterford, Blarney,, NewGrange, etc.)

We're doing the northwest coast in May, and we plan to include Galway, Clifden (because we're headed to Omey Island), Westport (for Connemara NP), Donegal, Dunfanaghy (for Glenveagh NP), Portrush (for Rathlin Island), and Belfast.

We spent 28 days in April, splitting the time between the North and the Republic. Also plan to go next September for 28 days again but plan on different areas. We have been several times and prefer basing ourselves in a self catering cottage for a week in 4 different areas. There are 3 of us with one that prefers to relax while the others explore so it's easier to not be driving as much. We spent our 1st week in Co. Down, staying at Narrow Water Castle, our 2nd in Gortin, Co. Tyrone (husband's family was from there). In 2 previous trips, we visited Derry (also family related) and loved it.
This time, our 3rd week was outside Galway. Since it was a longer drive from Tyrone, we stayed a night at the Slieve League B&B which was perfect. We were able to walk up to the cliffs from the house and had it all to ourselves. Amazing. We also were there on a weekend and went to mass-not realizing it was all in Gaelic! While in Galway, we visited Ashford Castle and had a session at the Falconry School. Also, a day spent on Inis Mor with electric bikes was very enjoyable. Our last week, we spent in a tiny town called Clonmellon in Westmeath. The owners of our Airbnb arranged for us to have a full day of fly fishing, something my husband has always wanted to learn. I tried to make sure we had a lot of different experiences but still had time to relax. We rented a car the whole time (I was the sole driver and learned I cannot drive a manual shift) which gave us a lot flexibility. However, our first trip there, we relied on trains and buses which I enjoyed. In September, we will visit, Co. Waterford, Cork, Dingle, and Wicklow. I'm in planning mode now. Have fun!

If you're interested in off the beaten path excursions and lesser known attractions, particularly for Connemara, Sligo, and Donegal , I suggest that you pick up a copy of "Backroads Ireland" from DK publishing. You'll find used copies on Amazon for just a few dollars.
Also, picking up your rental car in Dublin and returning it there would be your most cost-effective and convenient option The rental car concessions at the airport open as early as 5:30 AM, allowing you to get an early start on your day 3. A drive along the Old Military road thru the Wicklow Mountains would be memorable on a sunny day, and arriving at the Glendalough monastic site before the tour buses start to arrive at 9:30 is highly recommended - it's a whole different experience when you have the place all to yourselves.

I'm a bit confused... Are you intending to do "Wed 24 May -Wed 31 May: exploring West Cork, ROK, Dingle Peninsula, maybe drive up to Castle Island (wife's family is from Cahersiveen and Castle Island)" all from a base in Cork? If so this won't work as the driving distances are too great. Don't underestimate how long it will take to get between places. Driving in Ireland is slow! It is also somwewhere to slow down and enjoy. You need to relocate. You could manage from say Killarney, although that most definitely isn't 'off-the-beaten-track.' Look for somewhere smaller in that area. Kenmare might work, although it is further for DIngle.

If you drive to Donegal, you have chance to see something of the Burren (bare limestone scenery and quite unlike anywhere else in Ireland) and Galway which is delightful. It is less busy than Kerry but to mind scenically better. Try and build in time for a visit to the Aran Islands too. Further north there are places like Ceidhe Fields in Mayo which is possibly one of teh largest stone age settlements ever found, although most of it is now covered by peat. There is also the megalithic cemerery at Carrowkeel in Sligo and Creevykeel.

Your itinerary also misses two of the most important early Christian sites in Ireland - Glendalough and Clonmacnoise.

Don't stay in Coleraine unless you absolutely can't swing accommodation out at the coast. My favorite hotel is Bayview Hotel in Port Ballintrae which just might be worth the splurge! The view out over the small harbor is amazing and it is walking distance (since you like walks) to both Dunluce Castle and the Giants Causeway.

There is a bus that you take right at the Coleraine Train station that will go to Port Rush, Port Ballintrae, Bushmills, Giants Causeway, etc. There is no reason to spend two nights in Coleraine if you can be on the beautiful coast with a short bus ride.

Thank you everyone - it's helping me start to narrow down my options and I'm starting to feel less overwhelmed! @Robert, we're taking your advice on renting the car in Dublin. We decided to rent and return the car in Dublin and forego the train/plane altogether, I think it will be easier in the long run. I'll be back with another draft as we've added and deleted a few things. Also, found the Fleadh Nua festival in Ennis!