10 Days in Ireland

In 2018 we spent our last night in Dublin at a Holiday Inn Express near the airport. Nothing fancy; clean, efficient; what you'd expect from a HI Express. Next door was a larger HI Hotel; not sure of the name. They shared a large shuttle bus to/from the airport; reliable service.

Bonus: around the corner was a small, Italian restaurant we enjoyed: Little Venice

We like the Jury Inn in Galway. Everything we want to do is within walking distance, including a boat tour past the Cliffs of Moher and to one of the Aran Islands.

It'll be a great trip no matter how you organize the details, but here's my 2 Euro cents:
Given the high cost of rental cars at the moment you might research ways to limit the time that you actually need one. With that in mind I suggest heading to Galway after your time in Dublin - either by train or bus. You could then explore the town one day (without the need of a car) and maybe another day could take the ferry or fly over to Inishmore - a day trip to the Aran Isands would be a great outing for you, and you wouldn't need a car for that either since the ferry offers a shuttle (for a fee) from Galway to the ferry landing at Rossaveel, and a taxi would be pretty cheap to get to the airfield if you elect to fly.
You could then collect your rental heading out of Galway for the drive down the coast towards Dingle.
You can pretty much cover what there is to see in Dingle town and Slea Head drive in a day, so my sdvice would be to find a place to stay between Dingle and Killarney so you could cover both Slea Head and the northern part of the Ring of Kerry from a single location. We usually base ourselves somewhere between Glenbeigh and Castlemaine for that reason.
The one-way drop charges for rentals in Ireland are typically pretty cheap - cheaper than a day's rental these days anyway - so renting in Galway and dropping back in Dublin at the airport might be your more cost-effective alternative. If you didn't want to mess with the long drive back to Dublin you could drop the car near Killarney - there are a couple of outlets at Kerry Airport just outside of town for example - and then catch the train back to Dublin for your last night. Ryan Air has a couple of cheap flights per day from Kerry to Dublin if you prefer - for $23 last I looked. Though you'll pay extra for ala carte items like luggage check it'll still be cheaper & quicker than the drive back.
Many here (us too) rely on AutoEurope for bookings in Ireland - competitive rates and clearly stated options for the insurance cover.

In Dublin: the Merrion Hotel, convenient to everything and impeccable service, or the Wilder, a little further away (but still walking distance) from things like Trinity College, St Stephen's Green and Grafton Street but the nearby pubs and restaurants are less tourist-centric. Comfortable eccentric rooms with very good service. Haven't stayed in Galway or Dingle in years.

As for car rentals, if you are going to Dingle from Dublin consider a rental agency on the South Circular Road in Dublin. A number of Hertz franchises located there, including Conn's Ireland. Renting from the South Circular Road avoids having to go north before going south and west. It's an easy drive out of Dublin from there. You can drop the car off at the airport upon your return to Dublin. No additional fee from Conn's. For your last night before leaving consider staying in Trim (30-45 minutes to airport, Trim Castle, abbey ruins, Boyne River) or Malahide (10-15 minutes to airport, Malahide Castle, beach, excellent restaurants).