Itinerary help and best place to stay in the middle of our trip.

Do you plan to rent a car or will you be using public transportation?

We will rent a car.

A little minor surgery on your proposed itinerary would allow you to make the most of your limited time.
I'd start by jumping on the express bus to Galway upon arrival at Dublin airport, enjoy the city for a night (or two) and then rent your car there once you've had a chance to shed some of your jet lag.
With a car you could then explore some of Connemara to the NW of Galway, and/or meander south towards Dingle and the ROK. Unless you particularly wanted to drive back to Dublin you could return your car at the airport in Killarney and take the train back into the city. Fact is, you don't really have enough time to see everything on your list unless you're willing to spend most of your trip in the car.

Thank you, I appreciate your tips.

I agree with Robert's recommendation to rent the car in Galway. But with only five days at most, I'd visit either Connemara and the Cliffs of Moher (and perhaps one of the Aran Islands) or the Dingle Peninsula and the ROK. You can take a ferry to the Aran Islands from either Rossaveel (west of Galway) or Doolin. You'll enjoy a much richer experience in those places than at the bland Rock of Cashel and Blarney Castle. Killarney National Park, however, is spectacular and perhaps best saved for a future trip.

Thank you. I know we've packed a fair bit in, I was hoping we could see as much as possible. But, I do want to enjoy our time here as well. We do have 6 full days and then the evening we get in. I will relook at what we think is really important.

By heading directly to Galway upon arrival you could explore Connemara & maybe the Aran Islands for a couple of your days, then drive down to Dingle (about 3 hours by car) for another couple of days before catching the train back to Dublin for your last couple of nights as previously mentioned. That would give you time to enjoy at least a few of the destinations on your list without wearing yourselves out in the process.
For a first time visitor I'd say that one full day in Dublin city would likely satisfy your curiosity about the place. Popular side trips to Glendalough, the Wicklow Mountains and the Boyne Valley could wait for another visit. I'd sacrifice the extra day day in Dublin for an extra day in either Galway or Dingle in a heartbeat.
Book a car with automatic transmission - the default rental in Ireland is a stick shifter (and a left-handed one at that), and spring for the zero deductible insurance cover. Many here swear by AutoEurope.

We get in at 5:00 PM, I wasn't really keen on then traveling to Galway. Thinking of staying in Dublin first two nights, which gives us one full day in the city.
Then if we rethink our plans: should we home base in Galway and do a day in Galway, a day exploring Connemara/Abbey, a day down to the cliffs of Moher and the Burren. That still leaves another two days is going to Dingle and then back to Dublin by the airport for our last night too much? If so, what do we do instead of Dingle?

OK, first 2 nights in Dublin and the last one at the airport, which basically leaves you 4 more nights (or 3 full days) of touring. Given those constraints I'd suggest just staying in or near Galway and doing day trips from there. You'll be able to proceed at a more relaxed pace which will make for a much more pleasant experience for the 2 of you I think. The extra day might come in handy as a backup in case you catch a rainy day too.
Rather than basing yourselves in Galway, which we thought was pretty charmless as Irish cities go, consider the pretty village of Clifden about an hour NW of there from which you could spend a relaxing day driving the Connemara Loop and another taking the ferry to the Aran Islands from Rossaveal. Another highlight of the loop drive is Ashford Castle in the village of Cong. Could have tea or lunch at the castle, and the Hawk Walk there draws rave reviews from everyone who has done it.
My other suggested base would be the village of Doolin which would be handy for visits to the Cliffs and the Burren. Rather than battling the crowds at the Moher Visitor Center you might consider a late afternoon boat ride from Doolin Harbor to view the cliffs from a different perspective - the setting sun shining on the cliffs makes for a terrific photo op.
In June the days will be long - I don't think the sun sets until after 10 PM, so you'll have plenty of daylight for your explorations. If you elect to drive back to Dublin that last night you could stop by the monastic settlement at Clonmacnoise on your way - it's a very evocative place and is worth the short detour.