I think your ideas are generally good, Newgrange is slightly out of your way and I must admit I was underwhelmed by it but if you want to see a Neolithic tomb, and I am not one to stop you, this is probably the best. I must admit I cannot see the attraction of Stonehenge either but it doesn't stop thousands of others visiting!
The drive to Galway is around three hours, I would suggest a tour of Connemara on day 4 rather than the Burren, you could drive across the Burren, via the Cliffs of Moher if you like on your journey south to Killarney which to my mind has more to offer than Kenmare. To keep the journey as short as possible, you can take a ferry across the River Shannon to avoid having to drive via Limerick and that way you would just need accommodation in Dublin, Galway and Killarney.
The Ring of Kerry is famous, not only for the views but also for the traffic jams. I enjoyed the route but then on that day I wasn't driving. Many suggest the Dingle peninsular as an alternative but I haven't tried that so cannot comment. There is plenty to occupy your time in Killarney, both in the centre of town and close by such as Muckross House. the Gap of Dunloe and the Lakes.
Your journey back to Dublin from Killarney will be the longest of the tour and clearly will have to be taken the night before your flight home. You could of course stay in the centre of Dublin again but I would suggest one the coastal towns such as Howth or Malahide for your last night, lots of restaurants to chose from and a very short drive in the morning to return the car.