Hi, To answer your question: It is possible to leave Dublin on Tuesday morning, see Kilkenny and Rock of Cashel and still drive to Kenmare for the night (arriving late), though are you set on that itinerary? If you are locked into Kenmare and Ring of Kerry, I would choose Kilkenny OR Rock of Cashel (and maybe you can make a stop at Cahir Castle - one of my favorites -- also after Rock of Cashel). If you are not locked into Kenmare, I don't think you need to drive that far south since your priority is Galway/Ballyvaughan. You can also vary the route somewhat and stop and gain more out of the visit by not venturing as far south and doing the Dingle Peninsula rather than Cliffs of Moher given your time.
How about this:
Monday, Aug 13 - Land in Dublin and stay in Dublin - I suggest Hop-on Hop-off Bus to get around, making stops at Trinity College for a student-led tour and Guinness world if you are so inclined. I like the General Post Office exhibit and Kilmainham Gaol to give you a sense of some of the history of the country before venturing out of Dublin.
Tuesday, August 14 - pick up rental car and head toward Tralee - Stop at Rock of Dunamase in Portlaoise (google Rock of Cashel vs Rock of Dunamase); King John's Castle in Limerick; and the town of Adare - called one of Ireland's cutest villages. Stay in or near Tralee.
Wednesday, August 15 - Drive the Dingle Peninsula. Stay in or near Tralee.
Thursday, August 16 - Drive to your location in or near Ballyvaughn - I recommend crossing the Tarbery-Killimer ferry and driving up the coast to the Cliffs of Moher, then to your destination
Friday, August 17 - Stay in Ballyvaughn or in the area to see family that evening - I recommend driving through the Burren, stopping at some of the prehistoric sites and seeing the one-of-a kind landscape.
Saturday, August 18 - You could spend the day in Galway then drive back to Dublin Airport area. I don't think you will have worthwhile time to venture out to Connemara area then drive to Dublin.
Sunday, August 19 Fly out of Dublin for a 9:30 am flight
Alternatively, you could just go from Dublin to your location near Ballyvaughn or Galway and leave that as your base for the whole stay, taking day trips to Cliffs of Moher, Connemara, etc. Maybe stay a second night in Dublin. If yo don't want to explore the city more, take a day trip to Newgrange or Wicklow.
You'll have a great time, I am sure, whatever you decide to do.