Yes it's realistic. I drove from Belfast to Dingle in 5 hours in November of 2016. Got my license when LBJ was in office. Have driven the entire island (Eire and NI) twice. Once solo and in a car with a clutch. Being on the other side of the road has never bothered me. I use a paper map and now with goggle satellite view available plot my routes the night before making a note of towns between where I'm leaving and where I'm going. Helps if a wrong turn is made.
Dublin to Rock of Cashel will be 2 hours. Leave Dublin early to arrive at the Rock of Cashel when it opens. From there to Tralee another 2-3 hours. Will depend on traffic and weather. So I'm guessing you are staying in Tralee overnight? Seems a bit odd. Why aren't you driving from Cashel to Dingle? I'd opt for 3 nights in Dingle so you can comfortably see Killareny National Park, the Gap of Dunloe, Portmagee, Sneem, and Slea Head. To me this area is the most beautiful part of Eire.
Stay in Sligo. Shorter distance to Londonderry. Yes you can do Derry to Antrim to Belfast. At least one night in Belfast I hope if only for the Titanic Museum and the seafood chowder at Mournes by Kelly's Cellar pub!
I also hope while you are in Dublin you are going to see Newgrange to the north and Glendalough to the south? Easy Day trips. Like any city traffic sucks so keep that in mind if you have a car in Dublin. Make sure your accommodation provides parking. If you are renting after Dublin, rent at the airport as its easy to leave from there.
Lastly, remember Ireland and N. Ireland are two separate countries so you'll need Euros in Eire and Pounds in NI.