Have you already booked your flights? If not, look into flying into or out of Dublin and then into or out of Shannon. Would save you some time and you could stay on the west coast at the end of your trip instead of driving back to Dublin.
Rock of Cashel isn't really on the way to Kilkenny. Maybe consider doing that on your way to Kinsale, and stop at Glendalough on your way to Kilkenny? I think you could easily do Rock of Cashel and the distillery on your way to Kinsale.
Driving the Ring of Kerry and heading to Dingle is a lot of driving for one day. It's doable, but maybe consider cutting a day somewhere else (see below re: Londonderry). Also, you are really only giving yourself one day in Dingle. Although it is a small town, you'll want at least a half a day (in my opinion) to drive the Slea Head Road, and ideally more time to relax and enjoy the town and area. It was our favorite part of Ireland.
Galway to Londonderry by Google Maps is 4 hours one way, and Londonderry to Dublin is listed as 3 hours - and I promise you it'll take you longer than that. I don't think it is worth the time it would take you to get there to go.
So if you have to fly into and out of Dublin, maybe do:
5/16- Arrive in Dublin mid morning
5/17- Dublin
5/18- Dublin
5/19-Leave Dublin. Drive to Kilkenny with Glendalough along the way.
5/20- Kilkenny
5/21- Drive to Kinsale with stops at Rock of Cashel and Midleton Distillery along the way.
5/22- Kinsale
5/23- Drive to Kenmare with stop in the Killarney National Park along way.
5/24- Drive Ring of Kerry, stay night at Kenmare
5/25- drive to Dingle
5/26 - Dingle
5/27 - Drive to Galway with stops in Limerick and along Cliffs of Moher (this is a full day - you'll get to Galway around dinner time). Also consider skipping Limerick (a relatively unimpressive town) and taking the Tarbert-Killimer ferry to save time. From the Cliffs, drive through the Burren on your way to Galway).
5/28- Galway (Aran Islands trip)
5/29- drive near Athlone/Tullamore - go to D.E.W. Heritage site (you have expressed an interest in distilleries) and see Clonmacnoise (impressive ancient ruins). Stay around there (this is in between Galway and Dublin).
5/30- Back to Dublin, but stop at Newgrange en route
5/31- Fly out
Your itinerary overall is very good, the above is merely my suggestion. But I would strongly discourage you from trying to squeeze in Londonderry.